When I was 15-16, I played in a restaurant. I had an hour's worth of material, but that's all I'd do is one hour a week. Come back the next week and play the same thing. I wish I would've learned to do what aj talks about though. One night, I showed up to play and there was this other pianist who was playing. I felt bad for the people eating and listening that I had to take over. He couldn't read sheet music and only improvised. It was good stuff and much better than my prepared repertoire.
Another thing I did in high school was accompany the choir. I didn't get paid for that, only got credit. What it did though was put me in a position where everyone knew I was the pianist, and I was asked to accompany many of the band members and vocalists when they competed at regionals and stuff like that. They would pay me to accompany them, and I enjoyed doing it, so it was great. At one point, I taught lessons to kids in the neighborhood, but only a couple and I didn't charge much.
There are ways to make money while playing piano, but I also had normal teenage jobs and loved those, too. It's great to get out there and learn how the working world works. Have a manager, clock in on time, learn to work with others and deal with rude people, etc. As a mom, I have to advise you not to argue with your mother! She cares about you and wants you to be successful in life, however she frames the "argument". Most teenagers I know don't work, but my parents made me, like yours. I'm glad they did. Not only did I learn to work, I learned how to manage money, how to fill out my tax papers, how to save up for something I really wanted, how to open a bank account, etc. If you can find something that involves playing the piano, that is best (if you know that's what you want to do in life), but in the meantime, making money in any way is worthwhile. Well, except pimping - don't really do that, please.

Another thought that came to me is that there are companies that hire out performers for events. So, instead of trying to get your own name out there (which could be difficult at first), you can gain experience working for an established agency. When I was married, I hired a guitarist that way. I paid him directly, but I don't know how the funds worked out between him and his employer.
Anyway, good luck finding something you love!
