While it's quite possible that the liability portion of your policy would provide sufficient coverage for damage you may cause in an accident, your parents may be subject to "vicarious liability" since they technically own the vehicle. This means that it is possible for them to be sued if you hurt someone else in a crash.
Since North Carolina does not fall into the No Fault state category, filing an injury claim is less likely to impact your future rates but that's not exactly guaranteed. You can always shop for a new policy down the road if you do see an increase, but since you weren't actually injured you could always discuss the details with your insurance company and they can advise you of your options.
In North Carolina, the reinsurance facility is basically a non-profit organization that allows insurance companies to pool the risk of high risk drivers to keep rates low for consumers and create less risk for insurance companies themselves. Since insurance rates in North Carolina are capped, an insurance provider can cede (transfer) your policy to the reinsurance facility so that they share the risk with all other companies in the organization and you still benefit from a fair price.