I'm a dad now (damn, i'm getting old), so I'll
try to put myself in your parents' shoes. If your parents already think that you're reckless then allowing you back on the hill this season might be a foregone conclusion.
HOWEVER,
I would give it about 2-3 weeks, then arrange a formal sit-down with them and tell them
1 - You were initially upset and disappointed about the new restrictions, but you decided to mull it over while you calmed down and started to heal
2 - You're a little bit hurt about being called reckless as you don't want to sustain major injuries any more than they want you to, but you can understand how it must look from their perspective.
3 - You agree that in light of what happened, slowing down makes a lot of sense but instead of a complete ban on snowboarding would they maybe consider limited activity with the following caveats:
A - You will limit your time on the hill. Maybe no more than half a day at a time and no more than a few times per month.
B - No more being reckless. You've learned that lesson the hard way and have no wish to hurt yourself that badly again (this might involve a discussion on what "reckless" means).
C - In addition to a helmet, you will wear one of these -
Icon Field Armor Stryker Vest (or something similar. I've worn them while snowboarding and they don't really interfere with range of motion and you can't feel it it all when you fall on your back)
4 - Tell them that you don't like being told what cannot do (you're a teen, and teens push limits sometimes... its a part of growing up), but you recognize that you are way more fortunate that most other kids in that you have the opportunity and ability to participate in snow sports and that you do not want to lose that privilege if you can help it.
5 - If taking a safety, first-aid, CPR, or avalanche class would help give them peace of mind or serve as evidence that you intend to take safety more seriously going forward, then you are eager to do so.
That is just about the best case you can make.