Potentially unpopular opinion I’m going with ESH. There is more than one therapist and there is more than one prescription. (Speaking from experience). It doesn’t seem like you want them to work; which I understand is part of being depressed; but you need to give treatments a fair shake.
Have you thought long term about an animal? Not every place will allow you to rent (especially if you’re going to college), Roommates might not be cool with it, I just got a $300 vet visit, etc. A pet will not fix your problems only you can. Unless you’re getting something in a cage, it’s also unreasonable to think you wouldn’t need some help from your parents if it’s a cat or dog or something.
Your parents suck for saying they’ll pretend to be depressed to get a pet. However it sounds like this may be the only way you can have one where you live, cause I can’t see any other reason to fake it; so I wonder if they’re thinking it’s a family pet. That way it’s their responsibility and not yours? They still suck though.