“When I did my taxes with Turbo Tax, I owe $400 dollars to MA. Is this right?”--> I guess It depends. UNLESS there is reciprocal state tax return agreement between MA and Ct, you, as a non-resident of CT , must file CT return on your CT source income. To claim your state tax credit paid to CT on your MA return, as a full year resident of MA, you should file your MA return; as long as your MA state tax liability is higher than that paid to CT, you are still subject to MA state tax.
“ Being a full time student, not working in MA, working CT, why do I owe MA?”--> As said above, as a full year resident of MA, you are liable for MA state tax on all your income including your CT source income.
“ My parents claim me because they paid for my schooling. Any idea what is going on here?”--->Though you are claimed as a dependent by your parents, as long as you carry a certain level of taxable income, you need to file your return to CT and MA.