In certain countries, a "third party" is a political party which is an alternative to the two "major" parties.
The idea of two "major" parties is not usually present in a parliamentary system, which depends on multiple political parties.
In the United States, the two major political parties are the Democratic Party and the Republican Party.
List of American "third parties"
(This is by no means a comprehensive list, as anyone can start a "political party," though they usually don't get ballot access.)