At this point I've got a bit of a track record. So people realize that when 'Weird Al' wants to go parody, it's not meant to make them look bad... it's meant to be a tribute.
My own personal tastes don't really have an effect on whether song is a parody target or not. But having said that, I try to pick songs that I actually like because I realize that I have to live with these songs for a long time, from when I'm working on them in the studio to possibly playing them onstage for the rest of my life. So I try not to pick songs that I know would drive me crazy.
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.
In the '80s, I was the only game in town, I was the only one getting that kind of exposure in any rotation on MTV. Now with internet culture it seems like everyone is doing music parodies. And they're not all good!
I do original songs in the style of other artists, where I try to learn all their musical idiosyncrasies and try to do something that sounds like them and yet is a bit more sick and twisted.