Can a full-time student claim exempt and not pay taxes on their income?

Students are not exempt from taxes. They have the same requirement as anyone else. The only exception would be if your only income was from interest or dividends, at which point (if you qualify as a dependent) your parents could include your income in their income, but it would still be taxed.

You will need to file a return if your gross income was at least $9,750.

If you were claimed as a dependent, you must file if you made over $5,950.