There are multiple separate issues.
Can you contribute to an HSA in your own name? (HSA's are like IRA's, every account has a single unique owner.) You can contribute to an HSA in your own name if you are covered by a qualifying HDHP (high deductible health insurance plan) and have no other insurance coverage. If your parents cover you, but it is not an HDHP, then you can't contribute to an HSA. If your parents cover you under a family HDHP, then you might be able to contribute to an HSA depending on other factors.
(Side note: You can OWN an HSA in your name, and use it to pay for medical expenses for yourself, a spouse and your dependents, even if you aren't eligible to make contributions. A person who is covered under an HDHP and opens an HSA, and later changes to a different health insurance policy, can still own and use the HSA for qualified medical expenses, they just can't make new contributions. )
Separately, your parents can only use their HSA to pay for your medical expenses if you are their dependent. When you are no longer their dependent, they can't use their HSA accounts for your expenses, or if they do, the withdrawals will be subject to income tax and a penalty. Note that at age 24, you can't be claimed by your parents as a dependent any longer unless you are disabled and unable to work, or are not disabled but have less than $4050 of taxable income.
Thirdly, regardless of what kind of health insurance coverage you have, you can't open an HSA if you can be claimed as a dependent by your parents. As mentioned, at age 24, you can't be claimed as a dependent unless your income is less than $4050.
