Swifty asks: who first called it "The Democrat Party?" He suggests maybe Joe Martin, Republican Speaker of the House But apparently it's earlier:
... Newt Gingrich did a lot to encourage the use of “Democrat Party” as a not-so-subtle form of denigration, but the practice began well before he arrived on the scene. During the Truman-Dewey presidential election campaign in 1946, for example, B. Carroll Reece, who was then chairman of the Republican National Committee, used the adjective “Democrat” as a weapon. (Truman, in turn, suggested calling the GOP the Publican Party, a reference to the tax collectors of the New Testament.)Source: Grammarphobia, though without any citation to sources.
So the use of the term “Democrat Party” is quite an old trick. In fact, researchers have found references from as far back as 1855, though at that time the term may have been inoffensive and not intended to show disrespect. The linguist Geoffrey Nunberg has cited negative references from the early 20th century, but he says the practice didn’t become “a Republican tic” until mid-century.