I am sorry to hear that your parents wouldn't buy you a dog or a cat.
I understand how you feel and I hope I have some good advice for you.
When I was young, in the day when I told them that I want a dog, I was really hardworking and obedient and in this way I've proved them that I can have care of it.
First of all, you shoud not argue with your parents and try to do the same thing that I've done.You need to explain to them calmly that you really want a dog or a cat.
Concluding, I'm sure it will work out and they will show understanding.
Please correct it and sorry for bad english.