Your question does not make sense, I think there are facts missing.
Your parents can't claim your child unless your child lived in their home more than half the nights of the year. You can't claim an unrelated person as a dependent unless they lived in your home the entire year.
If you and the other parent lived in separate homes from your parents and child, then its fine for your parents to claim their grandchild (if the child lived there more than half the year), and for you to claim your partner who lived with you all year in a different home.
But if you all lived together, things get very sticky, and if your partner lived in the same home as the child, they are "custodial" according to the IRS definition, even though you might have primary custody through the court. (The IRS does not look at court orders, only where the child actually physically lived.)
So more information from you may be needed to respond to your question.
