Start small, ask them if you can have a study corner down there , something that doesn't need to be built, just a desk, a chair, a book case and an electrical outlet.

i agree with you to you should have your own room. but it also depends how much work needs to be done in the basement to make it, i dont know how much money your dad makes but he probably doesnt want to do it because of how much money it will cost

my mom is already on board but my dad is apposed to it. I am almost fifteen years old and I have to share my room with my sister that is five years younger than me and we are always arguing about things and I just need privacy now that im older and get a lot more homework that I need to be able to do without being interrupted

how do I convince them...

it would have to be built