Keep saving girls! Horses are a very expensive luxury, I work full time and still have be be extremely careful about how I spend my money with my horses. Buying the horse is the cheapest part, by the time you buy all of their tack, brushes, rugs, medical items plus farrier and so on, you will find that this will all add up so quickly that it will soon overtake how much you paid for your horse.
Saving towards lessons is a great place to start though, you've got a whole life of work to be able to afford your own horse ;) It is hard at 13 years old to find work, legally people cannot employ you. Would your parents be willing to give you some money for doing extra chores around the house than your normal chores? Maybe you could sit down with them and write up a list of what each chore is worth? Then it is a win-win situation for you and your family, you're earning some money, and your parents don't need to pick up after you where ever you go!!
Scrap metal, sounds dumb but good money for time invested. Lots gets thrown away. You can get it for free focus on aluminum soda cans and scrap brass and copper. Know any nearby shooting ranges ? You can get a whole bunch of brass quick. Get the old stuff not the stuff someone left behind.
Look for places that lots of soda and beer cans show up.
Ditto the babysitting! When my daughter was too young to ride with me, I had a teenage student come out and keep her occupied while I rode my training horses. I gave her free lessons in exchange. It worked out great for both of us!