I keep seeing posts like this. First off, either is correct as far as I am concerned, Democratic Party is the proper name for it, democrat party is just describing the party democrats belong to. It is a difference without a distinction. So what exactly do you accomplish by posting these ridiculous comments correcting other posters. Who appointed you editor of other peoples comments?
Also, if you do a google search for "Democrat Party" it suggests "Democratic Party" instead, so apparently Google agrees with the term "Democratic Party".