I think the advice YOU have been given thus far is good. Trying to talk your parents into changing their minds will likely backfire. You have to SHOW them and that means putting your head down and pushing forward with your goals.
That said, I personally think holding your drivers' license over your head is a terrible idea. I am an Eagle Scout, father of two Eagles and have been SM for 8 years. I've frequently advised parents against it. My biggest complaint is it pushes boys to try to earn Eagle earlier than they otherwise would and, in my opinion, should. The times I seen parents really stick to it, the kid earns Eagle at 15, gets his license and quits Scouting. That's not my goal. We work hard to keep Scouts engaged in the program all the way to 18. The game with drivers' licenses works against that.
Unfortunately, too many parents see Eagle as the purpose and goal of Scouting. It is not.