The child has to be a dependent on the tax return to pick up the Recovery Rebate Credit for the new child.

If parent #2 already filed the child as a dependent, then parent #1 should not unless parent #2 did so illegally. Then parent #1 would mail in the tax return and let the IRS reach out for proof of who had the child the majority of the time or any other tie-breakers for parents that do not agree.