That is the name of the party. Since 1828, America's oldest political party as founded by supporters of Andrew Jackson, has been known as the Democratic Party.
It is the official name of the party.
I know that. There are many of us who realize that the Democratic Party is anything but democratic. Using Democratic implies that they are democratic and they are far from being democratic. That’s why many of us don’t like the use of that term. It implies something that isn’t.
They are all Democrats. Therefore, it is the Democrat Party.
We call them whatever we want to call them.