Only one parent may claim the child care credit and that parent is the custodial parent.
You may be able to claim the Child care credit if you paid expenses for the care of a qualifying child to enable you to work or actively look for work. You can not claim the expenses that someone else has paid.
For more information on who can claim the Child Care Tax Credit see Publication 17 page 203.
Only the parent that lived with the child more than half the year (custodial parent) can claim the child care credit.
Child care is "work related". It can only be claimed if the care was necessary so that you (and your spouse if married) could work. Only if the child lived with you more than half the year *and* the care was required so that you could work is it allowed. If you did not live with the child or work at the time that the care was provided then it was not necessary so that you could work and therefore is not allowed.
See IRS Pub 503.