Research the kind of dog you want to get. Find pictures and information on how to care for it.
Make a PowerPoint presentation about the specific breed of dog and it's pros and cons.
Prove to your parents that you are responsible enough and that you are able to take care of the dog( I.e. walk it, feed it, clean up a mess it might make)
If your parents are allergic, there are breeds of dogs that are hypoallergenic
If your parents still refuse to get a dog, there are amny other options for pets such as cats, fish, birds, turtles, hamsters, guinea pigs