My username says it all.
Well, since I buy my cats’ food, serve it to them, clean their litter boxes, buy them new toys when they destroy the ones they have, supply scratching posts, play with them when they ask, and brush them, I’d have to say that I’m more an employee than an owner or parent. 
Seriously, I’m their owner. They’re great little companions, and I am glad to have them, but I’m not their parent.
I’m a doggy mommy.
Did you hit your dog’s car and have insurance? 
I am a proud father. My daughter is a 5 yr old German Shepherd named Hanna. And I love my daughter as if she were a dog.
They lived.
I was going to say something about this but decided not to… When we moved into a new house and got a dog all in one fell swoop my cats went off their food and two of them got hepatic lipidosis. One required 2 weeks in-hospital care at the vet, and the vet said that she had the worst blood work she’d ever seen a cat have–let alone a cat who lived. Both cats lived but it cost us about $3,500 and a feeding tube and all kinds of stress.
My cats lived without the drama.
I have 2 beagles and a cat. They are not picky about expensive food. If I buy expensive food they may eat or may not. They will gobble up Alpo. Any bowl of dry food will sit for about 3 days before it suddenly is gone.
My cat will eat Fancy Fest. Any other cat food is shoved on the floor. He eats pretty nearly any dry food. He just likes to change every once in a while.
Another owner who buys expensive super-premium food. I can afford it and anecdotally it seems their coats are shinier and softer and they seem to produce just a little bit less poop ( always a good thing in my book ). As I’ve mentioned before I previously had a cat for seventeen years who lived on mostly cheap cat food and did fine. But there IS a difference that is noticeable IMHO, particularly in coat quality.
Like NAF1138 said there is an element of selfishness at work there that works out for both sides.
And the many thousands I have spent on pet bills are arguably selfish as well - I’d be damned upset if I lost one of them ;). I’m very attached to my cats ( and before that the various dogs and cats I have owned ), but at the end of the day I try not to be overly sentimental or cloying about it. To me they are pets, not companions. Beloved pets, but still pets.
I guess I’d say owner or preferably, guardian. My pets are most definitely not my kids…not even when I didn’t have a kid.
My dog gets fed a homemade raw diet, but it’s not that much trouble and costs far less than premium food. My cats get whatever because they prefer cheap kibble to lovingly prepared fresh meat.
I’ll admit I do call the rabbits my “babies” when I’m talking to them, but I wouldn’t put them on the same level as children. (Then again I never wanted kids. 
)
I also get locally-grown hay for them via the woman who does rabbit boarding when we go on vacation - but that’s cheaper than pet store hay when you buy by the bale. (I bought what’s probably 3-4 months of hay from her for the cost of 2 weeks of hay from the pet store.) I also feed them any organic greens from our CSA farm deliveries that my husband and I don’t want - there are only so many salads or cooked greens (Swiss chard, turnip greens, etc.) that we can eat in a season. They usually get the equivalent of half a head of lettuce a day between the two of them for proper nutrition, so some of it is storebought “boring” greens.
Oh, and I coo over them, pet them, my heart melts when they come running up to me for petting or to play, etc. But I generally reject the label of “parent.”
Interesting question. I live alone and my pets are basically the only companionship I have. I call them my babies, and they definitely know who ‘Mom’ is. I buy them good food, treats and toys. I teach them right from wrong. I spoil them. They sleep with me. I tell them I love them regularly. They get regular veterinary care.
Still, they are not human. I am fully aware of this. I call them my ‘family’ but I know we are not related by blood. I love them like I have never loved anyone else in the world.
If anyone has a problem with that they can kiss my @$$.
owner, I don’t mind people who refer to themselves as pet ‘parents’, however I have to resist the urge to backhand people who refer to me or Mrs. Guest as pet parents or caretakers or whatever.
Ditto what rhubarbarin said about cat training, excepting the occasional personality quirk on the part of the cats I’ve owned over the years.
Neither. I’m a mom to my kid and I own my car. My dog and I share space.
(And I fork over for the kibble, the vet, the squeaky toys.)
I’m neither owner nor parent. They are my animals and I am their human. I feed them a major market food that keeps their digestive systems functioning at peak efficiency. The dogs have an alarming number of toys. The cat has an alarming number of cardboard boxes, bags, and crumpled pieces of paper.
If I had human children, I’d treat them pretty much the same.
We call our dogs “The Kidz” and refer to ourselves as Mom and Dad when we’re talking to them. Outside the house we don’t say any of this for obvious reasons…
They used to be on super-premium kibble until it became too expensive. The least offensive retail kibble I’ve found, for them at least, is the Rachael Ray brand. I personally never watch Rachael, but she’s lent her name to a pretty good product.
When I’ve had a pet, I’ve thought of myself as a pet owner, but with the understanding that owning a living creature is very different from owning, say, a house or a car. “Caretaker” might be a better term, since it implies a fundamental responsibility. But parent? No. I am a parent and my relationship with animals has never been anything near as important as my relationship with my sons.
Is “roommate” an option? 
I’m another one who doesn’t really like it when people refer to me as my cats’ “daddy.” (I had a landlady who would do this.) But I don’t really think of myself as their “owner” either. They are companions, not property. I believe I have an obligation to take care of them in a way I don’t have an obligation to my car. At the same time, they are ultimately their own masters.
When someone asks if the cats can be let out, I tell them that they are grown cats and can go out if they want to. That’s not entirely true–I don’t have a problem with keeping them inside at times, or with forcing them into a carrier to go to the vet, or punishing them for bad behavior. But at the same time, I really don’t feel comfortable forcing them to live inside permanently, even though I know it is safer. They are independent creatures with their own wills, and while I can try to train them to stay inside and make inside as appealing for them as possible, they have their own desires, instincts, and needs that seem to be satisfied best by being outside. (A dog would be different. A dog is a member of a pack, and within the pack it’s either the master or the slave. They like to be told what to do, and where to go.)
Both. I’m definitely their owner, but I also occasionally refer to them, tongue-in-cheek, as my furry babies, usually in the context of ‘who’s my furry baby den?’ while rubbing the dog’s belly.
My vet not only refers to me as my dog’s mummy, but refers to the dog as the cat’s big sister. But there are similarities in their relationship, I guess.