I'm sure they have plenty of confidence in you; otherwise they wouldn't let you race at all. And it kind of makes sense that they want to be there in case of an accident. It sounds like they just care about you lot.
If I were you, I'd thank them. Seriously. I'd tell them somthing like ...
"Mom, Dad, thanks for supporting my biking and driving me to races. You give a lot of time and money to allow me to enjoy the sport that I love. And even though get frustrated and think you're being over protective at times, I know it's just because you care about me and are being good parents. Not every kid has that, and I appreciate it a lot.
I promise that if you would reconsider and allow me to drive to the race with (whoever your friend is), I'll ride extra carefully. In the very unlikely event that I get hurt (your friend) can drive me to get medical treatment. And if you give me a copy of the insurance card, I can even make sure to get the treatment from the right provider. The point is, you're raising me to be a responsible young adult, and thanks to you, I'm becoming a responsible young adult. I would like to opportunity to ride both races. But it's entirely your decision, and whatever you decide, I respect it and am thankful to you for all you do."
Then, go outside and discreetly throw up. Just get these sappy words out of your mouth and into their heads. And after you say this, shut up and wait for their answer. Patiently. Don't hound them. It's your best chance of getting to ride both races.
Good luck.