To be gramatically correct, wouldn't it have to be 'The Democrats' Party'?, not 'Democrat Party'?

Use of 'Democrat Party' as an irritant goes back a ways-
Harold Stassen, commenting on the Wilkie-Roosevelt contest, said, "In the Wilkie campaign of 1940, I emphasized that the party controlled in large measure at that time by Hague in New Jersey, Pendergast in Missouri and Kelly Nash in Chicago should not be called a 'Democratic party.' It should be called the 'Democrat Party'...."
From an old Safire column in the NY Times.