Major Political Parties in the United Nations Parliamentary Assembly:
Global Democratic Party: The main big centrist-liberal party. Existed throughout the entirety of the history of the Parliamentary Assembly. "The natural party of government". Not the only party to become the "government" but is the one that did become "government" the most often. Had some policy changes throughout its long history. The main party supporting global unification. Later also growing to support unification of all of humanity.
Socialist Peoples Party/Federation of Social Reformists: The main left wing party in the UNPA. Mostly social-democratic throughout its existence but also had democratic-socialist phases. As long lived as the GDP. Also supportive of global unification (and later all of humanity). Generally less popular than GDP but has had times while they were the government. SPP would be replaced by FSR during the 1st Silent War. This party supported extensive reform such as abolition of vestigial organs that remained from the times when the UN was more Supranational Organization than a Sovereign Superstate, such as the Security Council and General Assembly. FSR also supported the creation of a new assembly as an upper house to the Parliamentary Assembly's lower house.
Jaamat-e-Islami/Democracy and Development/Party of Faith and Justice: The Islamic parties. Muslims have been either a large portion of the population of the areas under direct UN administration (I don't have the numbers but even if they were not majority they were definitely a plurality at some points). Tended to be the most socially conservative parties. The Jaamat was the earliest which functioned until the 3rd World War (2111-2114). It was succeeded by Democracy and Development. D&D tended to be more liberal than the Jaamat. It remained the big tent Islamic party until the 4th Interwar Era. During which it slowly lost popularity and was replaced by Party of Faith and Justice. PFJ was the Islamic Party operating during the end of the UN.
Alliance for Restoration of Independence/Sovereignty League: Big tent alliance of localist parties that want the end of "temporary" (that fiction was actually mostly done away with by 23rd century and was completely gone by the 24th) UN rule. Tended to be conservative. Sometimes grew popular but was mostly rather fringe. Was dead after the 1st Solar War and had grown very ephemeral almost a century before that. Spent the majority of the 77 years that was the 4th Interwar era out of the Parliamentary Assembly. Sovereignty League was created as a successor to ARI sometime after the 1st Solar War but was never very popular.
Legions of Gaia/New Legionary Party: Martian economist (from European Federation colonies, of Ukrainian descent) Alexander Kravchenko wrote and published "On planetary sovereignty and self sufficiency" in 2255. It was the birth of the Legionary Ideologies. The Book suggested that all planets should be self sufficient independent states and supposed that planets would be isolationist following this and systemwide peace would be achieved. As unification of Earth under its leadership was already an important part of UN policy by that time the original Legions of Gaia found an accepting home. They proved quite popular in the Parliamentary Assembly. Up until 2289 and the start of the 1st Colonial War this did not have many effects but with the Legionary revolutions in the colonies LoG grew to be shunted. The party managed to survive all the way until 2392 and the start of the 1st Solar War. As Legionary States were crushed across the Solar System the Legions of Gaia were banned. However the Legions would come back as the New Legionary Party after the UN defeat in the 2nd Colonial War (2444-2453). NLP would enter the UNPA after the 3rd Orbital War (2461) and would grow during the 1st Silent War, eventually becoming the government. Policy would change from the unification of humanity to only global unification and abandonment of all UN territories outside of Earth. NLP government also quickly began pushing for the replacement of the Charter (still in use, with changes) with a new constitution. In 2496, after a referendum, the Legionary backed constitution came to effect. The United Nations was abolished along with all its organs. The Parliamentary Assembly remained but lost much power. A more stratocratic model was put in effect.
Major Political Parties in the State of Gaia:
Legions of Gaia: NLP was immediately reformed into the new Legions of Gaia. It remained the government of the new State of Gaia throughout its history. In 2498 the 1st Silent War ended. As the Great Legionary Rising or the 4th World War started. The war was a defeat and the Legions were soon banned again.
Global Democratic Party:
Federation of Social Reformists: Both parties continued to exist during the State of Gaia and advocated return to representative democracy. There wasn't time for elections in the short existence of the State of Gaia.
Major Political Parties in the Earth Republic (Before Hermetic War) (Belém/Dakar):
Global Democratic Party: The old GDP still going strong. Party of liberal democracy fighting hard to maintain power. Still mainly centrist. And still going for global unification.
Alliance of Socialist Labor: FSR successor. Mostly moderate left. Quite popular. Unlike UNPA's mostly dominant party system, ASL got in power as often as GDP.
Islamic and Christian Democratic Party: Fairly similar to early 21st century christian democrats but in an alliance with Muslims of similar ideology.
Major Political Parties in the Earth Republic (Port Antarctic):
Nonpartisan: Operated as an Emergency Military Government from its "foundation" after the end of the Hermetic War for almost a decade.
Justice Party: Created by the military government. Mostly center-right. While it wasn't the only party in its parliament it heavily dominated it. Never lost the control of government until few years before unification with Earth Republic (Belém) and Earth Republic (Geneva).
Reform Party: The guys who managed to win the government and unified with the Belém and Geneva governments. Very big tent.
Major Political Parties in the Earth Republic (Belém):
Alliance of Socialist Labor: Direct continuation of the pre-war ASL. Since Belém was the executive and Judiciary capital the president's party survived the war. It did move closer to the center in this period however.
Christian Democracy: ICDP's successor. Did quite well. Was the government the most often along with ASL.
South American League: An unique party to say the least. Advocated for the end of the Belém government of the Earth Republic and replacing it with a pan-south american/caribbean/central american federation.
Major Political Parties in the Earth Republic (Kinshasa):
Nonpartisan: Operated under temporary military rule for quite a while like the Antarctic government.
Federalist Party: Dominant party. Generally right wing. Collapsed after the Great African War.
Peoples for Democracy: The general opposition. But somewhat more left leaning than the Reform Party in the Antarctic Government. Became the dominant party doing the governing for the Central African Federation, which succeeded the Kinshasa Government of the Earth Republic. But later joined the Belém/Geneva Government.
Major Political Parties in the Earth Republic (Dakar):
Global Democratic Party: Dakar was the legislative capital and as such all the major pre-war parties survived. GDP remained broadly similar to its pre-war version but moved towards the right slightly. Clearly less popular than the pre-war GDP however (probably less because of the party itself and more because of demographics).
Alliance of Socialist Labor: Likewise, survived. Grew somewhat leftier. Still one of the two largest parties.
Islamic and Christian Democracy Party: Again, continuation of the pre-war party. Remained mostly the same. Can arguably be said to have changed places with the GDP when comparing post-war Dakar to pre-war Belém/Dakar. Main rival of the ASL.
New Republic Party: Small Government, Free Trade. Also supports the reestablishment of orbital infrastructure and the return of interplanetary voyages, which had grown rather limited in the Inner System during the Interregnum. Celebrated the re-establishment of contact with Venus and Mars following the 3rd Lunar War and the Venusian conquest of the Moon despite the nervousness created by the aggressiveness (and well, the imperialism) of the Venusian Empire.
Major Political Parties in the Earth Republic (Geneva):
New Social Democratic Labor Party: ASL successor but decidedly more social-democrat and less democratic-socialist. Can be said to be arguably the most popular party in the Geneva Government but Geneva tended to operate under coalition governments far more often than majority governments.
Global Democratic Party-Continuation: As the name implies, the orthodox continuation to the pre-war GDP. Economically and Socially Liberal.
Global Democratic Party-Reformed: Social Liberal but economically more socialist. Somewhat similar policies to Green politics.
Peoples Democracy Party: Continuation of the ICDP but decided to to drop the open religious terminology. Can be said to be the most conservative major party.
Major Political Parties in the Earth Republic (Belém/Geneva):
United Left Front: ASL from Belém + NSDLP from Geneva + elements of RP (Port Antarctic) and PfD (Kinshasa). Big tent left wing party. The largest party overall.
Union of Global Democrats: GDP-C and parts of GDP-R from Geneva, large parts of Justice Party (Port Antarctic) and National Party (successor to the Federalist Party in the Central African Federation). A big tent right wing party.
United Democratic Party: Christian Democracy from Belém, PDP from Geneva and elements of PfD from Kinshasa. Proceeded with the same policies and ideologies.
Democratic Reform Party: Majority of RP from Port Antarctic, elements of PfD from Kinshasa and majority of GDP-R from Geneva. Center left.
ALLIANCE: Elements of Justice Party (Port Antarctic), South American League from Belém, elements of National Party (Kinshasa) and elements of GDP-C from Geneva. Right wing and heavily Populist. Smallest of the major parties.
Major Political Parties in the Global Federation:
United Left Front: Continues to exist as a big tent left wing party. Remains one of the largest parties.
Progressive and Liberal Party: Direct successor to DRP but has had some UGD, UDP and ALLIANCE members joining. The main liberal party.
Global Democratic Party-Conservative: Combination of majority of UGD and UDP. And also large parts of ALLIANCE. The main conservative party.
Reborn Global Democratic Party: Main successor to ALLIANCE. But has had members originating from all parties. Doesn't really have policies other than populist pandering. Tends to be fairly centrist because of that.