James, you cannot convince them. This is THE most common question, children ask on yahoo. Your parents rule their home. What they want, GOES. You do not get a "vote". If you live with them, at best - it is a benevolent dictatorship - NOT a democracy.
If you want a dog - you need to LIVE on YOUR own, and make enough to afford a dog and be a legal adult (age 18) in order to sign contracts for adoption, or to buy a dog.
You can promise anything, but children rarely (IF EVER) stick to their promises, more than a month or two. Your parents have to FOOT the long term vet, food and ongoing medical bills (at least for monthly flea, tick & heartworm prevention & for yearly shots) for the dog and will have to exercise it, if and when you leave..... to go to college.
I DO NOT recommend Pit bulls or pit bull mixes for anybody. While they can SEEM SWEET, many come from fighting dog backgrounds or carry the genes for that. That is ALL they were BRED TO DO. Some counties within states, DO NOT allow you to OWN them (BY LAW). Many home owner insurance companies or renter's insurance (should you live in a rental unit (now or later) DO NOT allow ownership of said breed and will DROP YOUR COVERAGE and you will NOT (repeat NOT!) be able to get any other company to cover you, if they find out you have a PIT or PIT mix. And since so many people roll their CAR insurance & home owner insurance TOGETHER (since it is cheaper to do so) - this means NO CAR INSURANCE. Are you getting my DRIFT?
DROP the idea of a Pit Bull or Pit mix! It is not a good idea and it is not going to happen - and you are foolish to even suggest it since YOU don't KNOW the ramifications of OWNING ONE. This is the problem with CHILDREN - who cannot look past their OWN needs and wants. You do not know diddly, about SQUAT.
If you already have a dog - which it sounds like you do, BE SATISFIED with that. There is no guarantee two dogs, will get along. If you cannot EVEN HOUSEBREAK the dog you have, FORGET ABOUT IT!!!!!!!! Housebreaking is BASIC! If you cannot manage that, with all the training options available - you CERTAINLY do not need another dog! (You do not get to PASS GO!) You'll just have TWO DOGS peeing in the house!! (Fortunately, your mother isn't stupid.)
If or when, your mother wants a dog, she should get WHAT SHE WANTS, since she will be stuck with it! (However, NOBODY recommends a teacup chihuahua) - they are RIFE with health problems & she needs to GOOGLE => "teacup health problems" before ever getting any such thing. ONLY irresponsible and unethical backyard breeders produce "teacups" - which is a marketing PLOY and a misnomer. (It is another name for a SICK RUNT).