third party vs political party

third party and political party both are nouns.

Word	Noun	Adjective	Verb	Adverb
third party	Yes	No	No	No
political party	Yes	No	No	No

As nouns, political party is a hypernym of third party; that is, political party is a word with a broader meaning than third party:

third party: a political party organized in opposition to the major parties in a two-party system
political party: an organization to gain political power

Other hypernyms of third party include party.

third party (noun)	political party (noun)
a political party organized in opposition to the major parties in a two-party system	an organization to gain political power
someone other than the principals who are involved in a transaction