My process for the parodies is that I get an idea for a song and then get approval from the artist and then go in and record it and probably try to get it out as soon as possible.
It was difficult to get into my friends' rock bands when I was a teenager. They somehow didn't see the need for an accordion player. That's when I realized that I had to find my own path in life.
So that's why one of my rules of parody writing is that it's gotta be funny regardless of whether you know the source material. It has to work on its own merit.
Not any specific one, but I was a huge fan of Frank Jacobs, I guess he wrote the plurality of the song parodies for MAD, Sam Hart, a few others, but that was also where I was first exposed to the art form of song parodies.