Most credits are not refundable. Which means if you have no income, you cant take the credit. Non Refundable credits are deducted from taxes owed, so if you dont owe taxes or enough taxes the credit wont help you either.
The EIC is a refundable credit. That means if you earned income (on a sliding scale) you can take the credit even if your tax liability is zero.
Parents can claim their college aged students as dependents, even if the student worked if the student meets the criteria for dependents.