Actually, sixteen is the age I'm familiar with in most states where you can work without getting working papers. Fifteen is the age cut off for the forum. The OP seems to have a good grasp of the English language, something someone too young for a forum probably wouldn't have.
Also, she's not trying to force them. The title states "persuade". Totally different things.
You're being kind of harsh. If she can't work, for one reason or another, telling her to "get off her butt" won't fix it.
If I would've accepted "no" with my parents in regards to horses, there is no way I would be leasing, taking lessons, or basically doing anything with horses right now. I had a dream, I had passion, so I didn't take no for an answer. I persuaded. I nagged. I did what I had to do.
Parents are not the same as gods. Not even close. They are at their core, regular people. Just like me, or the OP, or you. Yes, you should respect them. Yes, you should value their opinion. No, you should not take their word as gospel.
I've seen this same attitude from the forum before, and it really bothers me. Some of the people are acting as if people under eighteen can not only do whatever they want if they ~try hard enough~, but are stupid and spoiled if they do anything less than climb a mountain for what they want. Because back in my day, we suffered! We worked! We never wanted nor asked nor did anything less than perfection!
So, OP. Do what you have to do for your dream. Get a job, if you possibly can- I don't know. Read up on how to write persuasive things- there are a lot of helpful techniques out there. I think you're probably plenty grateful for what you already have, but there is nothing wrong with wanting more out of life. Maybe you'll eventually be able to persuade your parents to get you a horse- probably after showing a whole lot of hard work and perseverance- and maybe you'll have to wait. But you won't know until you try.