Two things that have happened in recent years:
1) Generally people have become more easily offended, and along the same lines people can take themselves far too seriously. Even in times like this people can’t loosen up.
2) Matt Groening once said, the reason the Simpsons keep going, he keeps finding comic targets. And some of the best targets are people that can’t laugh at themselves.
As far as musicians go, Prince was a perfect example. Weird Al wanted to do a Prince spoof, and Prince put an injunction out and killed it dead. “How dare you make fun of me? I’m an artist! My work is sacred! I’m The Artist Formerly Known As The Artist Formerly Known As Prince!”
Other musicians, are highly flattered when Weird Al parodies them. That’s a sign that you made it!