The U.S. has the Republican Party, the Democratic Party, and a smattering of smaller parties (including the Libertarian Party and the Green Party). The UK has the Conservative Party and the Labour Party. Germany has the SPD and the CDU/CSU (and smaller parties).
Has there ever been a transnational party which tried to get its members elected in different countries? I'm not referring to friendly relations between two parties in separate countries. I am specifically asking if there has been a single party, with a defined party platform, that has worked to elect representatives in more than one country.
Yes, there are several examples, both presently and throughout history. A great current day example is Sinn Féin, which is active in both the Republic of Ireland and in Northern Ireland (part of the UK). Sinn Féin has elected members in the Republic's parliament, the Northern Ireland Assembly, the UK's House of Commons (national parliament), and the EU parliament, although they refuse to take their seats in the British House of Commons. They are a unified party (as opposed to being two separate/distinct parties sharing a name, as you specifically are not asking about) and contest elections throughout all of Ireland, under a platform of Irish unification and republicanism.
Technically, the former KPD (Communist Party of Germany) existed in both East and West Germany for several years. The party existed prior to the separation in the immediate post-war period, with two branches becoming separate entities in each of the new countries. The West German KPD was later banned, while the eastern branch merged with the eastern branch of the SPD (Social Democratic Party of Germany) to form the single party SED (Socialist Unity Party of Germany). The SED went on to be the ruling party of East Germany, with a successor party existing today in reunified Germany (die Linke, the Left Party).
20 year rule.
So going back before that the Natural Law Party. Added an interesting flair to early and mid 90s UK elections. Ran candidates in rather a lot of countries but didn't have much electorial sucess. Got a fair bit of attention in the UK at least due to number of candidates and yogic flying. Transcendental Meditation and its claimed effects were a core part of their policies.