First question, have you parents seen you working at the facility?
Next, I would type up a proposal. Research the cost of feed, vet visits, farrier, supplies, board, etc. Break it all down for them on how you plan on paying for it all. Make phone calls if you have to and get pricing qoutes. Figure up a good amount to put aside for an emergency, cause trust me, they happen. Get a letter from the BO about how reliable you are, your skills, etc. Set up a time line with dates of when you would like to have stuff done by, such as in 2 months have $1000 saved for emergency. As a parent, I am more impressed with my children when they bring a me a well thoughtout plan then when they are just "whining" about how they want this or that. If you do this, hopefully, you parents will realize that you have thought about every aspect of horse ownership and be more willing to work with you. Let me know how it all works out. I actually did this with my mom when I was 14 and wanted my own dog and she didn't want me to get one cause I already had a horse. Once I showed her that I was willing to accept all financial responsibility, called vets about pricing, etc, she was impressed and let me get my own dog. Later, when I got married and wanted to take her with me, my mom refused to let her go cause she had fallen in love with her too.