You have to appeal to their desires. First off, make sure the only reason they are saying no is because of their worry of your safety. Let's assume for now, though I doubt it, that it's the only reason. If you can find out more reasons, you can take a similar approach though.
So, how can you convince them that you're responsible enough to keep yourself safe.
Do your research. Find out what kind of bike you want, make sure it is a reasonable size (don't try to get a huge bike just for your ego), find out how much it will cost, and how you or they or both are going to pay for it.
Where will you ride it? On streets with traffic? In the woods? On a dirt bike course? Tell your parents your plans. Research helmets and let them know you have one picked out that has a good rating. How about protective gear? Tell them you want to get a padded bike jacket to help protect yourself.
Do you know how to ride? Doesn't matter. What matters is whether they think you know how to ride. Tell them how you are going to learn, who is going to teach you, and how they are going to teach you.
Be thorough, and let them know you are serious about this. "I want it" is not a good reason. "I want it and here's why I think it will be ok...." is much more convincing. Don't whine, don't beg. Any objection they make, listen, and go back and figure out real solutions, and present them.
This is big boy negotiation. But it ALWAYS works IF you can figure out the real reasons behind their objections :)