COLGeek :
I would suggest a mature, unemotional conversation with your parents to understand their concerns. They must have a reason.
Seeking advice from total strangers on a tech forum website is otherwise not a solution.
Good luck.
Just what COLGeek said.
In all honesty, this isn't the place to ask.
We have no idea what your parents are like and what their priorities are. This is a tech forum dedicated to tech problems, not relationships.
But since you asked, I hope I can still help:
The best way to get a graphics card or any computer component isn't to "convince" your parents by pleading, rather ask them if YOU can work for it by making your own money. I'm still pretty young, and I have gotten some really nice computer parts like a 2k overclockable monitor and a mechanical keyboard by working and using my own money. I was even able to get my $1000 rig as a gift from my parents when I was 15. Remember, you serve your parents until your out of the house, so give them a good reason to give you an RX 480.