Have you thought about not dieting, but exercising more instead? Parents get funny about diets because they worry that it's going to lead to eating disorders, but I don't know of any parents who get upset about their kids taking up a sport or exercise.
I agree with the people telling you to talk to your Doctor though - s/he will give you an honest opinion and help your parents understand that trying to look after yourself isn't a bad thing.
Well done you for taking steps to address this though. :)