First: No, I don't think political parties should be protected by the First Amendment if they are organized as limited-liability corporations (C-corps, S-corps, LLCs, etc. -- unencorporated partnerships, on the other hand, are just fine).
Second: Political parties are, in general, a cancer and government recognition of them should be abolished. Closed primaries, faction-based voting and Gerrymandering are all violations of democratic principles caused by the existence of political parties, and the extremism & gridlock in Congress is an excellent example of the harm that results.
Third: Nothing I've said would in any way whatsoever harm like-minded people's ability to get together and advocate some political ideology. Said group neither needs nor deserves special recognition under the law that is not afforded to individuals, however!