Possibly. Let's say the parents live apart. Only the custodial parent, the parent whom the child spent more than 1/2 the nights during the year, can claim the Child and Dependent Care Credit. If the custodial parent passes the dependency exemption to the noncustodial parent, the noncustodial parent can claim the exemption and the Child Tax Credit, but not the Child Care Credit. The custodial parent can claim the Child Care Credit if they meet all the other requirements for being eligible for that credit.
If both parents live together with the child, only the parent who claims the child as a dependent can claim the Child Care Credit.