My parents claim me, and they live in DE. I am in college in Tenn., have a a part time job and all $ from my part time job was earned here. Do I pay DE state taxes?

Yes. Even though the income was earned in Tennessee, where there is no income tax requirement, the permanent resident state is allowed to tax all income earned with a credit allowed only for any amount of tax paid to another state. 

See 2019 Delaware Form 200-01 Instructions, page 5 stating that the income must begin with the Federal Adjusted Gross Income and page 6 about line 10, Other State Tax Credit.

@mattmoshe

Yes. A  student, attending school out-of-state, is considered a resident of the state where his parents live, if he is their dependent.