It sounds like your child is what the IRS calls a "qualifying child" of each parent. In this situation, the IRS "tiebreaker" rule is that the parent with the higher AGI (adjusted gross income) gets to claim the child.
https://www.irs.gov/pub/irs-news/fs-05-07.pdf
Also, if you provided more than half the cost of maintaining the household for yourself and a qualifying person, then you can file as Head of Household. See this:
https://turbotax.intuit.com/tax-tips/family/guide-to-filing-taxes-as-head-of-household/L4Nx6DYu9
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
