The political party Sinn Féin is active in both the Republic of Ireland and Northern Ireland. Of course, the desire of Sinn Féin is for a United Ireland, in which Northern Ireland becomes part of the Republic of Ireland. Nevertheless, it appears unusual that a single political party is active in more than one country.
Are there any other notable examples of political parties that participate in elections (at any level) in multiple countries?
There is a little difficulty working out what exactly is a political party, in a transnational context. The expectation is that a leader of a political party will become national leader, if they win an election. If the leader is from a different country it is likely to be a constitutional issue. For this reason, major parties are unlikely to be transnational.
Parties of the Right tend to be nationalist. I couldn't find any Right wing transnational political groups.
However there is the "Pirate Party". By its nature, the pirate party is not highly organised. There are pirate parties in various countries, they have aligned policies and membership of transnational groups such as "Pirate Parties International"(PPI). By some standards, the various national pirate parties are actually sub-groups of a single transnational party. However PPI this could be seen as an association of parties
Similarly various Socialist groups see themselves as local implementations of international communism. There is the "Party of the European Left" that represents a wide range of socialist and communist parties. They have aligned policies so by some standards the "Party of the European Left" really is a single party, and not an association of national parties.
It is a very reasonable to say that these are not actually political parties. However for comparison, consider the Conservative Party before the 1980s in the UK. There was no list of Members of the Conservative Party. There were local Conservative and Unionist Associations, and there was the Parlimentary Party (which selected the Leader by consensus rather than by vote) If we can accept the Tory party of the 1970s as a National political party, perhaps we can accept the Pirates or the European Left as transnational parties.
Ba'ath Party was the "Arab Socialst Movement" party and it was active in Egypt, Syria and Iraq.
Off the top of my head there are at least two mostly transnational parties in the EU:
A few more EP groups have transnational lines in practice if not on paper. Center-left in particular, aka socialists - as opposed to communists - in Europe.
In passing, there was a recent debate in the EU that revolved around whether to use transnational party lists rather than national ones for EP elections. Things didn't move forward, but you can tell based on who was for/against who is closer or further to being organized as a single party across the EU.
Besides that, look into parties related to ethnic groups that don't have a state and whose demographics span across multiple nations. For instance Kurds or Assyrians.
Some other examples --
Fenians in the 19th century. International Communist Party in the early 20th century. National Socialist Party from Germany in the latter 20th century.
National Rifle Association - while not exactly a party with a wide scope of policy proposals or pursuing governance, they are a lobby group that directly influences elections in the USA and Canada.
Jacobins in the 18th century - a political party during the French Revolution that sponsored American Patriots and participated in actions during the American Revolution.
Some Catalan and Basque political parties are active in Spain and France. For example, Basque Nationalist Party or Esquerra Republicana de Catalunya.
The Libertarian Party is active in quite a few countries.
The Pirate Party has an international presence.
The Green Party is active in quite a few countries.
If you consider the member nations of the United Kingdom to be distinct countries, then arguably all its major parties (ie, those who field enough candidates in enough constituencies) fit this bill.
While not strictly 'elected' in the same way as his immediate predecessor and successor were, Gordon Brown was a Scottish MP at the time of his run as Chancellor and as Prime Minister, as was the leader of the Liberal Democrats- Charles Kennedy- at the same time.
It may, however, be worth considering this a 'quirk' of the location, like with Sinn Féin, rather than a truly trans-national party, though.
