The short answer is NO.
In order to claim dependent care expenses, the taxpayer must also be able to claim the dependent on their tax return. IRS guidelines for claiming child or dependent care expenses limit this deduction to the custodial parent or taxpayer.
With respect to a joint custody agreement, the custodial parent is determined based on the number of nights the child stayed in the taxpayer's home. If the child spent an equal number of nights in each parent's home, then the parent with the higher gross income may claim dependent care expenses.
See IRS Publication 503 for complete information.
Note: This article answers a question about dependent care expenses and is not intended as tax advice.