1st look at your assets. Do you do your chores or have any? Do you help around the house? If so you have an advantage. Doing chores shows responsibility. Tell them you want a dog to help you be more responsible as an adult and future caretaker of others including your future children. If they give you that dog, then HOLD UP YOUR END by taking care of your pet. ENJOY.
A dog is a lot of responsibility. I would suggest bringing up the idea to them then showing them that you can be responsible and take care of things without being told.
Have you asked yet? Ask your Mom or Dad what you need to do. Ask if you get a dog what kind you will get, and what you have to do to keep it.
Since you do have other pets, maybe it is due to lack of space or money. If they are worried about space, there is nothing that you can do. If it is money they are worried about, work really hard to raise enough money to cover the fees of adopting one, vet fees and several months worth of food. You need to show them that you are capable of knowing how much a dog will cost and that you are willing to do whatever it takes to make sure that they don't have to shoulder the burden of paying for a dog.