There are several issues here. I know you are young, so I'm going to be honest, but friendly. Please don't take it the wrong way.
Issue #1 - If you aren't motivated enough to use the search function, I'm not sure you are motivated enough to be serious about a dog. 1300+ past questions for "how can I convince my mom to let me get a dog", 200+ "how can I convince my dad to let me get a puppy", 400+ "how can I convince my mom to let me get a puppy", 1000+ for a general search of "convince parents puppy", and a WHOPPING 3300+ for "convince parents dog".
The biggest issue is that you are still young. Usually, the typical teenager is very very busy with school, friends, boyfriends/girlfriends, thinking about college, work, and sports/clubs. Are you sure that you are going to be ok with staying home 12 hours a day to be with your new puppy? And that you are willing to not go to college or to work for the next 10-20 years?
And who will be paying for the dog? Dogs from a responsible breeder will be $750-$2000. On top of that, they will cost you about $2000 the first year in set-up, vet bills, and upkeep. After that, I'd expect $100 a month. AND, you'll also need $2500 in the bank for emergencies. You can't assume that there won't be one - most dogs have at least 1 emergency visit in their lives.
So, to "convince" your parents, you need to show them that you've thought about all these things. If this were my child I'd want him/her to show me and discuss the following things with me.
- Who will pay for the dog's upkeep, including vet bills
- A set in stone schedule, and punishment if the schedule is not adhered to
- A plan for who will be caring for the dog if the child gets too busy
- A plan for who will be caring for the dog when the child leaves the house
- A plan for the dog when the child moves out
- A plan for the dog's puppy training, obedience training, and ongoing training
- A plan for what happens to the dog when the child "gets sick" of the dog
- A list of responsible rescue groups and/or good breeders
- A list of why the breeder is good
- An overview of the breed's inherited problems and how the breeder will avoid them
- A clear understanding that the dog is a 15-20 year committment and will be a FAMILY dog
If your FAMILY will not care for the dog, then I'm sorry, but you should not get a dog. Wait until you are on your own and have lots of time to commit to one. : )