When you all live together as a family but are not legally married, the child can only be claimed on ONE of your tax returns. You cannot "split" the child-related credits. Usually it is better for the one who earns more to claim the child and file as head of household and get the other child-related credits. The other parent should file Single and enter nothing about the child on their return.
If a return using the child's SSN has already been accepted by the IRS, the other return will be rejected when that parent tries to e-file. If the other parent is the one who should claim the child, then their return has to be filed by mail. Even if you amend, that other return has to be mailed.
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