Since you are not the one paying for the horse you are not the one budgeting. The fact that your "budgeting" list consists primarily of grooming supplies says you are not ready for horse ownership quite yet. Grooming supplies are insignificant to the cost of ownership. I would consider leasing as an option first in order to get a better indication of the costs and commitment needed for maintaining horses.
It also depends on the costs in your area and the availability of quality services in your area. The cheapest boarding barn, farrier, vet, equipment, etc. may not always be the highest quality and can cost you even more in the long run. The costs someone puts up here can be misleading since they vary so drastically from area to area. As someone else mentioned hay in their area costs $5/bale. In my area if hay cost that amount it usually moldy hay that would make your horse sick! Here hay costs around $15-25/bale for good quality hay not including the cost of delivery or gas/truck/trailer to pick it up yourself. A good saddle can cost you anywhere between $500-5000. If it is below that it is usually something in need of costly repairs in order to be safe and functional or cheap, crappy saddles made in India that should never touch the back of a horse nonetheless be sat on. The saddle must properly fit BOTH you and the horse. Poorly fitting saddles will cost you in veterinary services, training issues, chiropractic/massage services, etc. I had a poorly fitting saddle wind up giving me horrible hip and back pain. You must also purchase replacement tack in case something breaks (my first horse broke 2 $100 bridles the first week I brought her home). Farrier services can run from $45 for a basic barefoot trim up to $300 for specialty shoeing every 6-8weeks. If you have a horse with poor feet, supplementation and more frequent farrier work may be required. Each horse has unique dietary needs and require a ration balancer. Grain is usually about $15-35/bag. A 50lb bag can last for a 10 day supply up to a 50 day supply as some can be fed at a rate of 1lb/day as others can be up to 5lb/day.
Vet services are pricy no matter where you are and accidents/illnesses happen no matter how safe an environment you try to provide for your horse. Surgery can costs up to several thousands with colic surgeries being about $10,000 minimum not including hospitalization costs with a deposit of usually $5000 down. Also as mentioned before you should at least have the budget to pay for euthanasia and disposal should worst come to worst. I've seen euthanasia around here cost about $200 minimum with disposal being around $350-400. If your horse comes up lame one day diagnostics can easily run up into the thousands. I would highly recommend getting a PPE on any horse you decide to buy. I had a friend who wanted to buy a young prospect for eventing. On the flexion test it was determined that the horse had a subtle lameness. The seller said the horse was probably sore from being shod. My friend decided to do further diagnostics to determine if the lameness was something manageable or reversible and ended up finding out that the horse had bone chips in both front fetlocks and quite severe arthritic changes that would limit the horse from being able to withstand the demands of eventing. She ended up spending almost $1k in diagnostics for a horse she did not even buy. You must also have a plan if your horse becomes permanently unsound for riding.
Owning horses on your own property can be very expensive as well depending on the region you live in. If you want to start from scratch start-up costs can easily become astronomical or if you want to purchase an already established horse property that will be even more expensive. Paying for the mortgage is something you as a child can not even fathom. You must have access to water, electricity, etc. Either you will have to pay for city water or dig a well (and you must have access to a secondary water source if your well dries up or becomes contaminated). You also have to have a waste management and pasture conservation plan that is compliant with your particular district including the machinery to be able to upkeep the land such as tractors, mowers, etc. You're going to need a lot more stable supplies than just pitchforks and water buckets. You will also need at least two wheelbarrows (one for feeding one for manure), hardware, lighting, fencing, gates, insect/pest control, storage areas, sheds, shovels, hoses, cold weather supplies, pallets to store the hay, etc. You also have to take fireproofing and insurance into consideration. Labor is also a major factor in addition to costs. Do your parents want to be responsible for maintaining a horse property 365days/year? Do they want to wake up early mornings to feed, muck, do chores, etc. in addition to having to go to work to pay for everything? Do they also want to become involved in horses or is all of this solely for you? Will this be something they want to maintain long term if you go to college? Hope that whatever property you move to has good neighbors. Awful neighbors can easily ruin all of the heart, soul, investments you've put into your property (you know like the ones firing their guns near your horses or running a meth lab in their basement...
Bottom line...horses are expensive. Since you are not going to be the one paying your parents need to be fully aware of the costs horse ownership entails if they intend to buy you a horse or purchase horse property.