You need to understand why your parents object to having a pet, then figure out a way to convince them otherwise. My suggestion is that your parents will get a pet, when they want it, not because you are asking. It would be better for everyone involved, including the pet if adopting it is your parents idea.
Consider the following.
1. Veterinarian bills are typically higher than going to a people doctor. So they may just be objecting to the cost?
2. Some people loath certain types of animals, cats for example. Cats need a litter box, which stinks and needs to be cleaned weekly. Cats have nasty habits, such as shredding your furniture or tearing up any house plants you have or male cats are known to "mark" everything with urine. If a cat happens to urinate on your carpet you may as well replace the carpet because you will never get the smell out. Some people just hate cats, or dogs or rodents...........
3. Children claim they will take care of "pet" they are begging for, but eventually (which may only take a few days) they start slacking off leaving the pet duties to the parent who never wanted the pet in the first place.
4. If your parents wanted a pet, they would have one.
5. If you travel at all, finding someone to take care of the pet is a royal PIA!!
Parent and pet owner.