They are entitled to claim you as a 'Qualifying Relative' if your income was under $4,050 and they provided more than half your support (TaxGuyBill already provided you with that answer). Otherwise, they are not. If they are entitled to claim you, you can not claim yourself as "independent" even if you wish to do so - to answer your question precisely.
As a practical matter, if you claim yourself, and your parents do not claim you (even if they are legally entitled to do so), the IRS will not investigate. However, if both you and your parents attempt to claim you, the IRS will investigate, and determine who is eligible to do so.
