It's an interesting linguistic dichotomy, the use of the two party names as nouns and adjectives.
In 'Republican Party' the 'Republican' is an adjective in the same way that 'Democratic' is in the phrase 'Democratic Party.' The similarity ends there, however, which is the source of so much deliciously-impotent Democrat rage.
'Republic' is an actual thing with its own distinct meaning, which is a type of national political system. The same is not true of 'Democrat,' which means an individual who believes in democracy and hence is the reason the current Democratic Party maintains the name (Ironically the party names are probably more appropriate than ever, since the Republicans believe in a strictly-chartered representational democracy while the Democrats seem committed to class warfare, division into interest groups, and ultimately mob rule).
A member of the Republican Party is therefore necessarily referred to in everyday language as a Republican, the adjectival form standing in for the more cumbersome phrase 'Member of the Republican Party,' where 'Member' is the operative noun and everything else modifies and classifies it, because it would make no sense to refer to him as a 'Republic' since he isn't an independent country with a voting population of one, electing himself to its own government for life.
Not true for the Democrats, however. "Member of the Democratic Party' is certainly just as cumbersome, but shortening it to 'Democratic' is just a non-starter, since the word is unambiguously an adjective in form, and therefore simply too grammatically out of place to stand in for the modified noun phrase. Amusingly (For us) the shortened form does transmute very easily back into an adjective, so that 'Democrat' means both a member of a certain Left-wing redistributionist party, but also of or pertaining to policies or things associated with that party, as distinct from the nonpartisan, classical meaning of the adjective 'Democratic.'
So, there is a clear linguistic reason why members of one party are 'Republicans' but members of the other are 'Democrats,' rather than Republics vs. Democrats, or Republicans vs. Democratics. There is also a clear linguistic reason why the policies or actions of one party are 'Republican' and those of the other party are 'Democrat,' rather than 'Republicanist' vs. 'Democratic.'
No charge for the lesson, lurking DUmmies. Consider it valuable citizen education gained while lurking at better-informed and better-written fora than DU, HuffPo, or the like.