Welcome to the horse world, and welcome to youth riding.
Regardless of your age, people will always question your choice at the mention of the word "horse". You absolutely do need a plan B. You never know what can happen that ends your riding career.
That being said, you're underage. Is it really fair to ask your parents to take on the expense of a horse ON TOP OF lessons? Because in order to keep progressing as a rider, you're going to still need lessons. As you know, horses aren't cheap, and once you own a horse, you can't just stop doing it tomorrow - still have to pay for the horse until you sell it.
On the up side, if they are willing to let you keep taking lessons, do so. Network. Make connections and friendships. Those connections are what will help you find a situation you can manage. Instead of just asking your parents to take on the financial burden of a horse - why not get a job and earn the money for more lessons or to cover the expenses of a horse? Your post kind of comes off as though you feel you should get a horse because you're serious and really want it. But things don't work that way. You're not entitled to it - no matter how serious you are, or how good you are. It has to be earned.
* I'm often reading and posting from mobile and Siri loves to make a mockery of the English language.