It sounds to me like you are choosing to take your stand on this, even at the loss of your permit. I think we're all in agreement about that.
Since you said the rest of your family is for your curls, could you talk to someone in your family who is close to your father, who he might listen to? It sounds like your mother is refusing to be a support in this, so can you find someone else who can be?
I also wanted to share with you something about me--when I was in high school, I was very ill and in treatment. My mother, who generally was the awesomist woman on earth to me, refused to admit I was sick. She wanted me to pretend the same. My grades fell (a lot), and I felt very alone. I didn't feel like I could tell any of my teachers what was going on, even though most of them asked (and some punished me) for my grades falling. I begged my parents for tutoring, which they blew off, saying I didn't need help and just needed to keep being the perfect student and child. So I kept it in, and I didn't get out of high school all that I wished I could.
I really regret not telling others and getting help. Your situation is similar in that I know that you really don't want to go against your parents in this, and you feel like you are disappointing them. It's probably making you feel miserable and like it's your fault, even though you didn't do anything wrong. One of the above posters is right, parents are sometimes wrong, and sometimes they need help seeing that. So, get help from your outside family, or even go to a trusted teacher or counselor if you need to do so. Perhaps even have them set up a mediation in this. Present all the evidence you can, about how it's hurting your self-image and how bad straightening chemicals are. But don't be afraid to ask someone for help like I was.
My father now can talk to me about when I was ill, and we can talk about it openly. He feels sorry and doesn't know why they didn't help me more. He regrets it. I think this is a situation where your father might regret his actions in the future, he just can't see that right now and might need some help to do so. It does unfairly leave the burden on you, but it's worth taking the steps to ask for help.
Averages out at 2b, ranges from 1c to 3a, very fine
Mane 'n Tail co-wash + leave-in
Clarifying shampoo every few weeks