They both sound like they are the work of a higher-planed yogi or shaman. Once, I asked a young hardware store clerk the location of chicken wire.
"You mean poultry netting," he said. He looked at me like an old woman who still called a refrigerator: ice box, a purse: pocketbook, or soft drinks: Coke. I still do that. It's a Southern thing.
One day "chicken wire" may become "poultry netting," but "pet parent" or "guardian" sticks to the back of my throat like how much I weigh.
Granted, the old title, "master," needs to be tossed out on the stoop, but I don't believe I devalue my pets when I say, "I own four dogs," or I call myself the owner.
There are probably those that believe the titles will remind people that our pets are part of our family and should be treated kindly. Or perhaps they hope the title will speak to owners who leave them outside or worse, chained. If it made a difference, maybe I'd say it.
If people don't take care of their pets because of their humanity, or even because it's the law, then I doubt what I call myself will matter.