@Lyndaj
Under no circumstances.
this thread is years old, and the 2019 date is probably just a date that it was transferred to the new board software. I did not review the entire thread for accuracy, just your question. However, in the case of unmarried parents living together and sharing custody, only one parent may claim the child and all the tax benefits of that child, the other parent can claim nothing. The special rules that allow the benefits of a dependent to be split between two parents only apply to children whose parents are divorced or separated and live apart and share custody.
In TurboTax, the parent who will claim the child will say that the child lived with them for more than half the year and that they do not have a custody agreement with the other parent. The custody agreement question only applies to a legal or court order between parents who are divorced or separated and does not apply to a voluntary understanding between parents who live together. The other parent should not list the child on their tax return at all.
In the case you described, the parent who paid more than half the household expenses could file as head of household by claiming the child but would not qualify for EIC because of their income. Or, the parent who paid less than half of the household expenses can file as single and claim the child as a dependent and possibly receive EIC, but not both.
