My parents are understandably upset and I could use some advice

Hi, I'm 16/F. I'm afraid I have to begin this post with a confession. Back during summer break, I was responsible for a car accident that severely injured two people in the other vehicle involved. I was lucky enough to walk away with some bruises, but they had to be taken to the hospital in an ambulance. The accident was a result of the fact that I'd been texting while driving and also speeding. Once the police finished their investigation, I was charged with two counts of vehicular assault. Fast forward to last Friday. I had to go to juvenile court to learn my fate. The judge sentenced me to three months in juvenile detention. The reason I'm not there right now is because I was shown a little bit of leniency. Since this is mid-terms week and since I've always been a good student, her Honor is allowing me to remain free until Friday evening. But once school ends that day and Christmas vacation begins, I have to report to the county juvenile detention center to begin serving my sentence. Christmas, New Year's, my seventeenth birthday...I'm gonna be spending them all in jail. I own the fact that what I did was stupid, reckless, and criminal. I feel absolutely horrible about what I've done and I know I deserve my punishment. My parents have been furious with me ever since the accident. I know I betrayed their trust horribly and I hate how strained our relationship is right now. I want to show them, not just tell them, how sorry I am and that I've learned from this mistake and will never be so stupid ever again. But how can I do that? Any advice anyone may have will be greatly appreciated. I need to start getting ready for school. I don't know if I'll get any replies to this post before tomorrow, but I promise I will read them all once I'm released. Thank you all for listening. Writing all this out has made me feel a little better, I think.

There is no quick fix here. It is going to take an extended period of time, in which you'll have to continually prove that you are acting more responsibly.