There could be primarily two reasons why your parents may not be keen on getting you a dog. It could be their apprehension of the discomfort they may face while having a pet at home. Otherwise, it could be could be their assumption that taking care of a pet might be too big a responsibility for you.
Regardless, if you need your parents to get you a pet, you will have to change their perceptions about you or about having a pet.
Take the first situation. Your parents may be worried that the dog will make the house messy. Or they may not be open to the idea of taking the dog for a walk everyday. Whatever it may be, please keep in mind that you are the one who wants to get a dog. You should address their concerns and convince them that you will take care of the pet and that it won't be a burden for them.
You should show your parents how badly you want to have a dog. You could also show them the benefits of getting a dog: that it would give them a sense of security; that it would contribute to your development as a person by imbibing in you the qualities of being responsible for something other than yourself. Once you have addressed all of such concerns, you have a better chance of convincing them to get you a dog.
There is always the possibility that your parents may be apprehensive that taking care of a dog would be too big a responsibility for you. You will have to convince your parents that you will do all things necessary to take care of the dog and that you will not get bored of the dog after a while. A good way to do so would be to assist them with the household chores to show them that you can handle responsibilities. Tell your parents that you will feed the dog, walk it everyday and that you will ensure that the dog does not turn your house upside down.
You may have to be a bit patient as it may take time to convince your parents that you are responsible enough. Your parents may also be worried that getting the breed of dog that you want might be expensive. If your parents have such a concern, you can assist them by saving up any pocket money that you get and giving the same to your parents. Trust me, that is one move which can express how badly you want a dog also also indicates to them that you are responsible enough to take care of the same.
Finally, once it all works out and your parents get you a dog, make sure you are good on your word, that you will ensure that your promises are kept.