Hey
Why dont you just go out and buy one without them knowing
? They wouldnt have much to say about it then. Unless of course they make you take it back. which is the worst part
. I used to have the same problem with my parents but lucky for me the arent as stubborn as some parents are.
That`s the only thing that I can suggest you do unless you tell them that it`s your money and you really want to play the guitar, and that you will do anything for them if they they let you get it. If that dont work just buy it. other wise your never gunna be able to buy one until you leave home.
Goodluck matey.