For at least 80 years, some Republicans have referred to their major opponents as members of the "Democrat Party," not by its name — the "Democratic Party." It's a jab suggesting that the party is not democratic.
We refer to the party by its name: the Democratic Party. Its elected officials are Democratic senators, governors, etc.
This is not new guidance.
Our style guide says "NPR uses Democratic" when naming the party.
William Safire wrote about it in 1984.