(((Princess))) I've been where you are and I know it's frustrating. Do not lose your dreams, don't let anyone tell you they are unrealistic.
At 13, you don't have lots of options. If you're living in a big city (which is what it sounds like) you have even fewer. You cannot get around your parents. At 13 they control your life and if you're planning on university, it sounds like it will be a few years until you have the funds and ability to be independent. Get your education, start earning some money and then do whatever you want.
Meanwhile, all is not lost. You don't have access to a real riding school, but you do have a kind lady who lets you ride for $15 an hour? WOW..
You do not need tall boots to ride. You need something with a heel and a helmet. Jeans aren't great, but you want to ride: LOOKING the part is secondary. What you do have is time. LOTS of time. And time is a precious gift. You know what you want to do...so start investigating.
If you have a public library, you have access to all kinds of horse information. If they dont' have the book you want, they can get it for you from any library in the US (If you're in the US) It's called Inter-library loan and it's usually free. You can use this time to educate yourself about horses: anatomy, care, feeding, first aid. There are also books on riding, and although learning to ride on a horse is far better, you can educate yourself about position, aids, etc. Get yourself a big round ball that people with bad backs sit on...Practice, practice, practice. Talk to the lady who lets you ride...she may be able to help. Practice using the aids on the horse you do ride. You have TIME on your side.
By the time you actually can afford to take real lessons, at least you'll have a working knowledge of terminology and the basics. You can't learn EVERYTHING from books...but you can get a good head start.
There's a lesson in this: You can't always get what you want when you want it. But with ingenuity and patience you can usually get there if you hold tight to your dreams. Sometimes you have to take a detour....but you get there in the end! Good luck!