You are eligible to contribute to your HSA so long as you are covered by an HDHP - whether yours or your parents.
Double covered doesn't matter so long as they are both HDHPs.
But if either plan becomes a non-HDHP, then make sure you get out of that plan before it starts on you. You can't contribute to an HSA if you are covered by your parents' non-HDHP (for example) even if you are covered by an HDHP at work.
