Well, your parents are correct in that it is not cheap to go racing. Besides the initial cost of the car you will need a way to get it to the races ( tow vehicle/ trailer), entry fees, repairs and maintenance costs, etc... Working and saving your money is your best option at this point. Doubtful that you will convince your parents at this point. It's hard to find a sponsor or sponsors without a ride. Not saying that it can't be done; But you are asking them to put up their money and you are risking nothing.
If your plan is to become involved in racing, other than as a driver, there are several options. You could volunteer at one of your local tracks as a corner worker, score keeper, etc... Any racers in your neighborhood? Offer to help around their shop or garage. I started out changing tires for my neighbor when I was a couple of years younger than you are. I helped friends work on their cars and volunteered my services as pit crew. Eventually, I got my first shot at racing in a mechanics race. Other options could include (after you get your drivers license and first car) racing in SCCA Solocross or Autocross or drag racing your street vehicle in high school drags or bracket classes.
There are options for you. You just have to create opportunities sometimes. If your passion is cars; learn to work on them (high school auto shop, welding class, whatever). Perhaps not the answer you were looking for, but it's somewhere to start.
Best Of Luck To You!
Steve