Since your parents are paying with money that has not been taxed by the U.S., you will not be able to claim the credit. The IRS states:
Expenses Cannot Be Paid with Tax-Free Funds
You cannot claim a credit for education expenses paid with tax-free funds. You must reduce the amount of expenses paid with tax-free grants, scholarships and fellowships and other tax-free education help.
Your answer from the IRS about limited is trying to explain that you may be able to get benefits up to the amount of your tax liability but nothing additional. The program asks many questions but to answer correctly you need a lot of background information. It sounds like you have done a lot of work!
AOTC is only good for 4 years and covers books and additional items so it is the better credit.
Reference: Publication 970, Tax Benefits for Education
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