"Apparently my parents are now not allowed to drive my car and I am not allowed to drive theirs. "
It's the car that is insured, not the driver.
Your mom does not pay so "she" can be insured, but her car.
So technically even if you where not insured you could drive her car legally. It's only her insurance for her car that matters.
Your mother/father can let ANYONE with a valid license drive there car, for as long and as many times as you want.
The idea is that if someone is REGULARLY using your car (say that you moved in with your parents and started driving there car PRIMARILY) AND this person using it does NOT have an insurance policy then they should be added.
Uninsured drivers can drive insured cars.
On the other hand if you've had your license taken away obviously you cannot drive any car, for any reason.
Not the most reliable source, but this "Google'd Up" resource agrees with me.