I would define being a parent as either having adopted a human being or having conceived a human being. Why do I care? I think words should have a useful meaning. If we broaden it to anyone who has a pet then we would require a new word for those that have a human child because in some cases that distinction is important because it is truly fundamentally different in several ways. If someone organized a local meetup group for new parents it should be understood that people who have recently adopted a cat would not be welcome. If the word parent is broadened in this way what would we then have to call that group, a "new human parent group"?

I can ask you: what are you missing in the term "cat owner" that you feel you need to use "cat parent" instead?