Do you already have other animals at home? If so, make sure to stay on top of their care. Fill water/food bowls when needed (without being asked), walk the dog (if you have a dog) or play with the cat (if you have one.) Proving to your parents that you're responsible enough for your own pet may be the ticket to convincing them.

If you have no pets in the house currently, I suggest volunteering at your local animal shelter, as well as doing research on the type of pet you want. That way you gain hands-on experience as well as some basic knowledge on animal care.

Also be sure to follow through with the responsibility of caring for your pet if you do get one! Don't let everything fall to your parents. If you work, buy the food and toys yourself. If you can, pay the vet bills (or at least split the cost with your parents.)

That's all I have for now. Good luck!