No, the parents would claim it all. Even if the student paid for it, the person claiming the student claims the expense.
Question 2. You may be able to take the Lifetime Learning Credit for your high school student, but not the American Opportunity Credit. In order to claim the AOTC you have to be enrolled in a degree program. Generally, the courses taken in high school are not part of a degree program. The lifetime credit does not have this stipulation. There is no longer a tuition and fees deduction.
https://www.irs.gov/pub/irs-pdf/p970.pdf
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