People have already given you some good advice. I personally didn't start riding until I turned 18 (6 years ago-literally the month I turned 18) and could afford to pay for lessons myself and not have to deal with any liability issues being underage. My parent's refused to pay for lessons for me, and have never helped me pay for any horsey things. They never had a ton of extra money, and they were the type that said I'd grow out of the horse crazy thing.
If stopping your lessons isn't totally necessary to their finances, what about if you start taking lessons every other week to cut the cost in half? And think about other ways you can help them with money. Any other activities that you're willing to give up that cost them money? Even stuff that uses up their vehicle's gas. Any clothes, electronics, etc that your parents buy you routinely that you can go without? I know 13 yrs old is a hard time to make money.
If your family really does need the lesson money- turning 16 really isn't that far away. And around here, you can get a real job at certain places when you turn 14 (like fast food restaurants). Being involved with horses is a lifelong learning experience and stopping lessons for a time won't affect that. Look around at other barns to see if they'll let you come hang around their horses/help with chores, which may or may not lead to riding time or lessons. Read and learn from books or online from souces like this forum. See if you can watch other people's lessons at your current barn. Watching lessons is a very good way of learning more. If it's within their means, ask for something horsey for you birthday and christmas. Like some lessons or a paying to go to a horse camp.
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