Insurance companies generally require any household member of driving age be listed on a policy. Policyholders normally have the option, however, to "exclude" anyone they don't want added to their policy. Unfortunately the accident you had is exactly the reason why insurers require household member to be included on a policy. In the event that you shared any fault in the accident then they would have to cover the damage. It is likely that the reason they aren't representing your parents is due to the fact that the other driver is 100% at fault. In the event that there is a dispute with the other driver's insurance company about covering all the damage, you may have to hire an attorney if your parent's company is unwilling to step in and assist.
If the only issue is that your parent's company is requiring you be added to the policy, your parents should have the option to exclude you so there is no impact to their rate. The 3 of you may also benefit from shopping around
with new insurance providers to see what else may be available. Ultimately we recommend shopping around every 6 months
to make sure you're getting the best deal.