No. In this situation only one parents may list the child as a dependent, and that parent can use the dependent care credit. The other parent should not even list the child on their tax return.
(There is a weakness in the Turbotax interview that can lead to both parents claiming tax benefits for the child, if you answer "yes" you have an agreement to share custody. This refers only to a court order between parents who live apart and share custody, and you need to answer no, there is no agreement in your case, to prevent the child from being claimed twice.)
