I have seen it brought up in many places, though obviously it is generally only Democrats who do so. But anyone paying attention to the news would, I would think, pick this up after a while if they were discerning and can make basic inferences. Republican politicians talking on TV refer to the “Democrat Party”, while Democratic politicians refer to the “Democratic Party”. (Meanwhile, both sides say “Republican Party”.) Since the people who say “Democrat Party” are not Democrats, and generally do not have kind things to say about Democratic policies, that ought to be a hint.
As to your allegation of hypocrisy, I dispute that. I do not deny Republicans the right to say “Democratic Party policies are terrible” or any variant of same. I only ask that they use the correct name for the party. Similarly, I have said that my impression is that the Deep South is not a desirable place to live. And yes: I even said that I have an impression of it being an “ignorant backwater”. But I have not used a name for the region that is not generally understood to be its actual name, in particular a name for the region that its residents pointedly avoid using.