As hard as that might be for you, yes you should. Provided she's a decent person and won't cause them any harm.
and pays child support each week
sounds like a "proceed with caution" situation to me...
You have the right as their mother to set boundaries. She does not have the right to discipline your kids or play parent in anyway. Setting boundaries is vital. These are YOUR children and you are their only mother, they must be your first priority and you DO have the right to say that his time with them should be without her if you have reason to feel that way.
If he's serious about her than yes. It eases the situation overall for your children to know who their father is seeing instead of keeping them in the dark--even if it is difficult for you
they are every bit as much his kids as your kids and you have no right to tell him how to live his life as long as the kids are in no danger it is not your choice anymore
that is his choice you do not have the right to tell him how to live that part of your life is over as long as the kids are in no danger it is not your choice if he plays house with his kids or not as much as that makes you mad it is the sad truth of the matter
actually, by law and sense that IS her right to set boundaries. Daddy's little gf doesnt have the right to parent her children.
I agree on the point of parenting but as far as even spending time ??
Have you met her? I would want to meet her first, then I would allow it, I would also think you would do the same for him.
Thats a reflection on him, not you, he shouldnt be letting antone around your kids unless he is sure he is going to let them stick around, I see where you are coming from now.