How did you get the cat in the first place? If your parents do not give you the dog you want, there is always a reason. You can always adopt one or buy one yourself and keep it at a friend's place. When possible, visit the dog.
You can walk your neighbor`s dogs. And you can meet your classmates who have dogs. You can have contact with nice dogs even if your parents don`t buy you one. And one day you can yourself buy your very own dog!