No surprise, my AP Stylebook presents it with some ambiguity. I'm thinking whomever wrote the article also posts for Dictionary.com:
"Capitalize both the name of the party and the word party if it is customarily used as part of the organization's proper name: the Democratic Party, the Republican Party.
"Capitalize Communist, Conservative, Democrat, Liberal, Republican, Socialist, etc., when they refer to a specific party or its members. Lowercase these words when they refer to the political philosophy.
"Examples: John Adams was a Federalist, but [one] who subscribed to his philosophy today would be described as a federalist. The liberal Republican senator and his Conservative Party colleague said they believe that democracy and communism are incompatible."