I stick with the principle to do what is best for the kids. If they want to go to a party arranged by their mom, then I have no problem. My X has been easy to work with and we have both cooperated in the best interest of the kids. I handle the Cub Scouts and Girl Scouts and when one of the events occur on her weekend, she is willing to try and take the kids to it. Sure there are schedule conflicts (like this coming weekend), but we work together to solve them without letting our personal feelings cause a conflict.
Do you feel that your STBXW is wanting to have them attend the cousin's BD party just to spite you or is she doing it because the kids will have a good time?