Assuming the child is under the age of 24 and does not provide over one-half of their own support, then the parent where the child spends over 183 nights of the year in their home can claim them as a dependent (temporary absence for education is considered living in the home). Only the parent claiming the child as a dependent can claim education expenses and credits on their tax return.
The parent who cannot claim the child as a dependent cannot claim any education expenses for the child on their tax return.
