There are no "European Parties" at least not how one generally understands the term in national politics. Candidates for the EU parliamentary election are elected on the ticket of national parties. Those parties are free to form alliances (on a national basis) but most don't.

Once a candidate is elected and joins the parliament, they usually join a faction of like-minded MPs from across the EU, i.e. conservatives, liberals, social democrats, greens. Those are NOT parties but function similar to i.e. the faction structure in German parliament which may also contain members from multiple parties. Politicians from different national parties may end up in the same faction in EU parliament but that is rare.

This is more or less the same how it works for national elections in many countries, it is called "representative democracy", simply adapted to make sure that the EU parliament is roughly representative of the EU electorate.