The other parent to claim the child tax credit

Yes, this is possible if you claim the child as Head of Household for EIC, but the other parent claims the child's dependency exemption and thus the $1,000 Child Tax Credit.

If the custodial parent signs a release, the non custodial parent will get exemption and child tax credit. Even after signing Form 8332, the custodial parent is head of household and get EIC and any child care credit.