Your HDHP coverage being provided by your parent's insurance plan does not disqualify you from being eligible to contribute to your own HSA. What can disqualify you is your parents being eligible to claim you as a dependent on their tax return (regardless of whether or not they actually claim you) for the year for which you would be making the contribution.
See IRS Pub 969 for the HSA eligibility requirements:
https://www.irs.gov/pub/irs-pdf/p969.pdf
