It's a little more complicated than that. The Democratic Party was founded around 1828 under that name, but in early use many people apparently used "Democrat party" as well, without prejudice.

Kalamazoo Gazette (Sept. 1, 1838), p.2:

"Resolved, that the allegation so often made and repeated by the Federal Press that the Democrat Party is opposed to the credit system, and to all Banks, and in favor of an exclusively metalic [sic] currency is utterly false."


It only became an obvious slight when the Democrats themselves quit using it, and the Republicans started using it all the time.

The invidious imputation may be that a "Democratic" party stands for democracy, but a "Democrat" party is just made up of people who call themselves small-d democrats.

If I had to guess, I'd guess that the Republicans began saying "Democrat Party" in earnest during the Depression, when they absolutely despised Franklin Roosevelt, but it would be only a guess.