No, PA does not have a reciprocal agreement with Delaware so you will have to pay the taxes to Delaware for the work performed in Delaware. You will get a credit on your PA resident return for the taxes paid to Delaware on this income so you will not be double-taxed.
Then why doesn't Turbo Tax show a refund from PA? It just shows what I owe DE but I should get a refund from PA correct?
Probably not. Pennsylvania has a flat tax so most people break even and get no refund or pay no tax (you still have to file).
You can claim a credit for tax paid to DE but only up to the amount of tax paid to PA. Your DE tax will cover your PA tax, but since it's DE tax, PA won't refund extra DE tax on the PA return because it's not PA money that was withheld.