PETA's goal is animal rights, not copyright maximalism
PETA is not seeking to extend copyright, they are attempt to extend animal rights.
PETA is a strange animal, if you'll excuse the language. Unlike, say, ASPCA, they are not really concerned about the welfare of animals; rather, they are concerned about whether animals should have legal rights.
I doubt PETA understands what copyright even is... They just want a monkey to "own" "intellectual property."