If you are on your parents insurance but they do not claim you as a dependent on their tax return then you can open an individual HSA in your name (and you should because your parents are not allowed to pay for your medical expenses from their HSA if you aren't a dependent).
But you can't open a family HSA because you are only the one you. Sorry.
As far as should you open a plan on your own or through your employer - the money is tax free either way and the hassle is less if it's done through a payroll withdrawal and shows up on your W2. So if everything else is equal do it through your employer.
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