You know you have an especially thin skin if you get all angry and emotional every time the President refers to the "Democrat Party." If it's a slur, it's an extremely petty one. More than twenty-five years ago, when I was working for a Democratic state legislator, we would "push poll" Republicans and identify our guy as the "Democrat candidate." It enabled us to inform the person at the other end of the phone that we were honest-to-goodness, rock-ribbed Republicans of the Bircher stripe, and thus made it easier to stick the shiv in when we asked if it would effect their vote if they knew the GOP candidate was a Klansman, or performed abortions, or whatever it was we wanted to communicate.
As Congressman Miller said, it's code, like flashing gang signals, that indicates the speaker is from the far right. It's not like being called the n-word, or using the word "Jew" as a verb or adjective, and the number of people who are so identified with that amorphous ideological blob known as the Democratic Party who could rightly take offense is so minuscule as to be irrelevant.