Dad would need you to sign an IRS form 8332 releasing the exemption to him to legally be able to claim them. Which is not to say he couldn't try to lie to his tax preparer....but that's a different story.
If the kids live primarily with you, the court order specifies that the exemption goes to you - then legally you are the one that claims the children.
Right, so OP doesn't need to do anything to stop it - just put the correct exemption information on her own return. Eventually, the IRS will discover that two people are claiming the same kids and investigate. If she's got the kids over 50% of the nights, then she is entitled to the exemption and Dad is going to end up paying back taxes and penalties.
As an aside, if the parents get along reasonably well, it is always worth calculating taxes both ways - Dad gets the exemption and Mom gets the exemption and then agreeing to what ever is the most practical.
For example, let's say Mom is CP and if we calculate taxes with Mom getting the exemption:
Mom gets $1,000 refund
Dad gets $0.00 refund
Now, if we calculate it the other way (give Dad the exemption):
Mom gets $0.00 refund
Dad gets $2,000 refund
So, even though Mom doesn't legally have to give Dad the exemption, they might work out a deal where Dad gives Mom $1,500 in exchange for her signing the 8332 form. Each parent would be $500 better off in that scenario.
This scenario is reasonably common when one parent makes considerably more than the other - OR in situations where CP makes enough that the deduction has phased out and isn't worth anything to them.
But if the two can't sit down and reasonably discuss it and be open about their finances, I'd probably just stick with the IRS rules unless there's a court order that says that CP has to sign an 8332.