I am a resident of Oregon and I am a dependent on my parents who are in Oregon. I study/work part-time in Pennsylvania. Do I owe state taxes to Oregon for income in Penn?

[TomD8]

Yes. Oregon taxes its residents on all their income, regardless of where it's earned. The income you earned in PA is also taxable by PA, but you can claim a credit on your OR tax return for the taxes paid to PA, so you won't be double-taxed.