I am 16, and when I bought him, I was 15, and he was bought in my name, and his papers are in my name. I didn't even bring a parent with me, my boss drove me there with her horse trailer. I will be sure to talk to her parents, but I will allow her to sign the horse in her name.
I bought horses like this myself, and sold them, signing for them in my name, before I was 18. That said, you don't have a valid contract. The primary 'signees' must be of legal age, meaning the buyer and seller. If you ever had to take action, using the bill of sale as a legal document, it would be invalid, thus completely nullifying the point of having it in the first place. Do ahead and negotiate price with the girl, but she should not be the only one signing(nor should you). My parents don't know the head from the tail, so I did all the negotiations, but they should have been there to sign(or even my BO).
Don't ever let the horse leave without being paid.
Don't ever take a persons word for it. Deposit in hand and everything in writing.
Don't ever assume it will work out based on phone conversations and emails. I've had the most perfect sounding buyers come look(several different cases) that swore up and down that the horse was perfect and they were showing up with the trailer, only to have them turn out horribly wrong for the horse, or not the way they portrayed themselves.