As a sort-of side note, if the parents don't live together, that changes things.

Only the Custodial Parent (the parent who has the child more nights during the year) can claim the Dependent Care Credit.  The Custodial Parent can claim that every year, regardless if they are claiming the child as a dependent or not, and the Noncustodial Parent can never claim the Child Care Credit.

The Noncustodial Parent can only claim the child as a dependent if the Custodial Parent gives the Noncustodial Parent Form 8332.