Should I file my own taxes if I'm a working college student who lives with my parents and they assist me financially?

If you are under the age of 24, a full time student and you do Not provide over one-half of your own support then your parent's can claim you as a dependent under the Qualifying Child rules if you meet all the requirements.

If you are claimed as a dependent then when you file your tax return make sure that you indicate on the return that you are a dependent.

To be a Qualifying Child -

1. The child must be your son, daughter, stepchild, foster child, brother, sister, half brother, half sister, stepbrother, stepsister, or a descendant of any of them.
2. The child must be (a) under age 19 at the end of the year, (b) under age 24 at the end of the year and a full-time student or (c) any age and permanently and totally disabled.
3. The child must have lived with you for more than half of the year. Temporary absences while away at college are considered living with you.
4. The child must not have provided more than half of his or her own support for the year.
5. If the child meets the rules to be a qualifying child of more than one person, you must be the person entitled to claim the child as a qualifying child.
6. The child must be a U.S. citizen or U.S., Canada or Mexico resident for some portion of the year.
7. The child must be younger than you unless disabled.

as to whether you need to file, you have not provided enough info so look at chart B and C in the 1040 instructions.  even if you don't have to file, you must file to get back any income taxes withheld.  state filing requirements differ from federal

https://www.irs.gov/pub/irs-pdf/i1040gi.pdf