I've never been sure why I'm supposed to find the term "Democrat Party" insulting, other than that Republicans pronounce it as if I should. Why does chopping off a syllable turn the word into a slur? And why doesn't countering it with "Republic Party" have the same impact?
One explanation, offered by linguist Geoffrey Nunberg, is that it avoids associating the Democrats with small-d democratic, a term with positive connotations: "The point of the maneuver was to suggest that there was nothing particularly democratic about a party whose support was based in urban political machines." (Nunberg suggests the term is "at least" as old as 1923. Wikipedia finds earlier uses of the expression.) With "Republic Party," on the other hand, you're replacing the neutral word Republican with the positive word Republic--not exactly an effective counter-maneuver.
I, however, have a different explanation. What Republicans are doing is using the noun Democrat in place of the adjective Democratic, thus vulgarizing the term. It's similar to why a phrase like "Jew doctor" (as opposed to "Jewish doctor") is considered offensive. Obviously, this method of creating an insulting term only works if the adjective and noun are different words. You can't pull this trick with Republican, which is both the adjective and the noun: people say "I'm a Republican who belongs to the Republican Party," as contrasted with "I'm a Democrat who belongs to the Democratic Party." Thus, "Republic Party" misses the point, because chopping off the final syllable of Democratic isn't in itself what makes "Democrat Party" insulting.
The third and final explanation I've heard, mentioned by Hendrik Hertzberg, is the most mundane: "Democrat Party" simply has the rhythms of a slur. Like other slurs, it is sharp and quick, the kind of word you can just spit out. The secondary stress gets placed on crat, which contains rat. We can't do that with "Republic Party": while it does contain ick, the stress is on pub. That would be a pretty clueless way of trying to insult Republicans unless we wished to insinuate that they go to British bars.