Of course it's not rude. Your MIL has a problem but that doesn't have to be your problem. If she had severe allergies, it would be rude to allow someone to bring their high-shedding dog. But to bring your parents and cousin?
The irony in your MIL's feeling is that Thanksgiving is about being grateful for what you have. So one would hope that she would be so happy that her son married a wonderful woman who is warm, welcoming, and honoring of family. She should be THRILLED that her son's chosen partner is someone who is giving, generous and organized enough to pull this off.
I think it's a terrible shame that your parents couldn't come to a once-a-year event like the Halloween parade! Your in-laws will stay with you, I presume, since they are from out of town, while your parents will return to their house. Your in-laws will be there for breakfast, lunch and dinner, while your parents will give them plenty of special time with the grandchildren. But to require or suggest that they skip Thanksgiving dinner or a parade or a child's concert or graduation? I'd put a stop to that right now and let your MIL deal with her insecurities on her own or with help from your FIL and your husband. There is NO reason to deprive your children of all those pictures of all 4 grandparents in the house, and all the memories, because your MIL hasn't developed the skill of the average 6 year old to be able to share!
I think, if part of the reason the 2 sides don't "mix fantastically" is that your family could be a little more warm, then you can try to urge them to take a little bit of a back seat and maybe drop a compliment or two about how nicely your kids are playing with the other grandparents or how well your IL raised your husband. Beyond that, I wouldn't enable a bunch of people who can't get along without hogging everything to themselves.