Yes, Democrats prefer to call their party the Democratic Party. It is reportedly understood by Democrats that to use the term “Democrat Party” is a minor epithet as per CatBus’ explanation of disrespect. NPR mentioned it in a segment a few years ago.
JEDIT: I found an even older article than what I was looking for: https://www.npr.org/sections/ombudsman/2010/03/since_when_did_it_become_the_d.html
That’s a bit of an unfortunate url, though.