Income paid to you in cash, or reported to you on 1099-MISC in box 7 is self-employment income. It gets reported on SCH C as a part of your personal 1040 tax return. The fact that you qualify as a dependent on your parents' tax return and they claim you as such, doesn't matter.
If you have self-employment income in excess of $400, then in addition to regular income tax, you will pay an additional 15.3% self-employment tax on the income over $400. That's basically the employer side of your social security and Medicare, and it gets credited to your SS/Medicare account.
