(Kidding, and congrats on the child)

I completely agree with you that being the parent to a human child is harder, more work, more stressful, and that the stakes are way higher. I can absolutely understand feeling belittled and upset if someone tries to argue that raising a child is anything like taking care of a pet. It's not.

I don't think that this is the argument that most people who refer to themselves as "pet parents" are trying to make, but certainly some of them might be doing so. I think that that's silly and easily refutable.

However...

This whole thing seems too similar to the religious conservative argument that referring to gay unions as marriages was degrading the sanctity of their own marriages because of a word. That argument has always seemed pretty pathetic to me. The weightiness of a concept isn't diminished when people decide to recycle a word [1]. The sanctity of someone's marriage isn't defined by use of the word marriage. Similarly, the sanctity of human parenthood has nothing to do with how dog owners want to refer to themselves.

Language evolves, and it's usually pointless to fight this process. Sure, ambiguities that are introduced may be annoying, and you may get frustrated that you now have to clarify something when previously you didn't, but there's no reason for you to have the right to dictate how other people decide to see themselves. Trying to fight this with outrage and offense seems to me like shouting at a rain cloud for getting you wet.

If people want to refer to themselves as pet parents because they don't currently have or aren't planning to have children ("No, I'm not a real parent, but I am a pet parent"), leave them be. If they seriously want to make the argument that taking care of a dog is anything like raising a kid for 20 years, shake your head in smug amusement and leave them be. They're obviously wrong, and you know it. Hell, I even know actual parents to human children who also have a dog or two, and who consider themselves both parents and pet parents. They're clearly just trying to be cute rather than equating the two. Enjoy it if you think it's cute, or don't, and then leave them be, because at the end of the day, it doesn't affect you. It'll only affect you if you let yourself get upset by it, and that's just a really poor use of your time and energy.

[1] Obviously a lot more was/is at stake for the LGBTQ community, and I don't mean to belittle actual LGBTQ struggles for equality and acceptance by comparing them to the plight of people who want to be cute by referring to themselves as pet parents (there, I guess even I need to clarify).