No, you cannot be your own dependent. That is, "claiming yourself"' only means your claiming your own exemption ($4050 deduction). A 'dependent" is someone in addition to yourself.
"So my parents could just had send the original tax form through paper and they would've gotten the tax breaks last year"? Yes. If you are talking about 2015, and they filed without claiming you, they can still file an amended 2015 return to claim you (assuming they meet the rules above). You would have to file an amended 2015 return unclaiming yourself.
