My suggestion would be to:
1) learn all you can about wet shaving from this site.
2) purchase shave soap,a brush and a bowl.
3) Work on shave prep while using a parent approved razor.
4) Ask for a straight razor for Christmas/Birthday.
If you show your folks that you are responsible enough to do all the homework involved in learning wet shaving, they might reconsider. Being patient and respectful will help them see that you also have the maturity needed.
Good luck!!!
p.s. With my folks,it was motorcycles. You will notice by my name, that worked out well. Thirty-seven years and still riding. ;)