Safety. The number one concern for most parents is safety, which is completely understandable. But what you can tell them is that, as with any sport, dirt bikes can be learned with a "crawl, walk, run" approach. Meaning there are plenty of small, very manageable "first bikes" on the market that make it very easy to dial down the power (list to follow). Also, even more performance-based kids' bikes have throttle limiters and basic tools to keep things as fast or slow as they want. Also on the safety front, you can show them all the gear on the market just for kids. Not only does it show them that you can be covered head to toe with every manner of protection, it also shows your parents that the motorcycle industry cares deeply about safety and that there are enough kids riding to support such a robust array of kids products.
Motorcycles are amazing tools to teach responsibility. Regarding riding, tell your parents that you will learn to be responsible for your own safety and learn to ride within your limits. Regarding your bike, there are many things that you need to check before you ride so, with their help, you will learn good safety habits and get into a routine of taking care of your bike.
You'll spend more time with your parents. Tell them that dirt bike riding is such a core family activity that you will beg them to either ride with you or for them to watch you ride. Families that ride together, stay together. This goes beyond just family as well. Some of our best friends and strongest relationships are with fellow riders who we've met along the way. There is a massive sense of community and belonging that is supportive and understanding.
Riding is healthy, especially off-road riding! There is a reason that motocross and off-road racers are in such good shape. Riding a dirt bike is a great physical activity that engages your whole body as well as your mind. Tell your parents that your reflexes will improve, you'll get plenty of exercise, and your mental acuity will be sharper (plus using fancy words like that will impress them).
You'll be too busy to get into trouble. Once you start riding, you won't want to stop and it will be part of most of your free time (after homework and chores, of course). Riding a bike or working on a bike are constructive activities that take time and keep you occupied and fulfilled to help you ignore bad influences as you grow.
Positive stress release. Even kids get stressed out from school work and peer pressure, but you can say that riding is an exciting stress reliever that lets you recharge and focus.
Vehicle control. Motorcyclist are always better drivers than non-motorcyclist. Tell your parents you will be a much safer and aware driver because riding a motorcycle teaches you to be focused and aware of your surroundings. This translates to driving a car when you get your driver's license.
If your parents are still not responding to all of the compelling reasons above, try to work out a deal.
Pay for the bike yourself. Come up with a list of chores you can do around the house and work on how much each pays with your parents. Pick some hard ones too. Sure, they are lame, but it will show your parents your dedication and willingness to be responsible.
Leverage your grades. If you have good grades, promise they will stay that way and you won't let them slip. If you don't have good grades, this is the perfect opportunity to impress your parents and get something rad out of the deal.
Read more. Work out a book-for-cash or book-for-riding time agreement. Not just schoolbooks, but reading all kinds of books is overall good for your brain. Parents love to see their kid reading rather than playing video games or on the computer.
Take more classes. No, not at school but at dirt bike riding schools. There are plenty of riding coaches and classes that even provide bikes, which is a good way to try before you buy. This shows that you are being responsible while learning good riding and safety habits.