NO. Only one parent may claim the child care credit and that parent is the custodial parent.
For tax purposes, there is no such thing as joint custody, regardless of what your legal agreement says. The requirement, to be custodial parent, is that the child live with you MORE than 50% of the time. One of you has to be the custodial parent and the other the non-custodial parent.
But, the custodial parent may count the money paid by the non-custodial parent in calculating the child care credit for herself. The general rule is that a taxpayer must actually make the payment (as well as be legally liable to pay it) to get the deduction. However, there is a recent tax court ruling that seems to say that you can take the deduction even if he was the one that actually paid it under the theory that his paying the expense is just another form of child support and it was your money that paid for it.
