Well, I bought my daughter her first horse when she was 10. I also bought myself one a few months later. You'll gather from that that I also ride and know a lot about horses (I actually got my first pony when I was 5, but I don't recommend it). So do I think a 13 year old can own a horse? Sure. However, you need to be in one of two situations for that to work. Either a) your parents are into horses as much as you, and are very experienced and knowledgeable (I didn't get that from your post) or b) your parents are willing to pay all the bills, including paying for full board and a coach to help you with the horse. Without a parental figure to guide you, it will be important that you continue to learn (it is a lifelong journey!), so you'll want to have a coach or trainer who can work with you and your horse. I'd also encourage you to spend lots of time with others who will be interacting with your horse, such as a farrier or trimmer, and the vet. Ask questions. Keep learning.
If you were to ask whether it's a good idea for a 13 year old to get a horse, bring it home to their backyard, and be solely responsible for it without any adult help, I would say no. And I say that from experience because I was put in that situation when I was only 11. I did my best, but did not do everything right. We have horses at home now, but as the adult, I take full responsibility for the horses. My daughter helps when I ask (she will be 13 in a couple of weeks), and if I need to be away for a few days she is in charge (with a lot of instructions, and pre-mixed feed, and help from her dad). But it is a huge responsibility for a 13-year old to do everything. Boarding, on the other hand, is totally doable. The horse's basic needs are being met. I know lots of girls that age who have their own horses. Heck, I know a girl that age who trained her own horse, and is about to train her second horse, an OTTB! She's pretty impressive, but also rides every single day, no matter the weather, and outside to boot! Her parents are also very committed, and help out at the boarding stable, drive her to shows, volunteer at the stable. But I'd say she's the exception, not the rule.
Just make sure your parents know the full cost, and are willing to commit to it long-term. If you want to convince them, you might put together a budget that includes initial purchase cost (horse + tack + blankets + brushes, etc.), the boarding cost, the cost of continued lessons, farrier/trimming costs, regular veterinary cost and the potential cost of emergencies, trailering and show costs if you're going to show, etc. etc.