A.V.
I would tell your sister (for example), "Remember when we were all kids and Mom would take one kid with her and how you got some quality time that way? It's kind of like that. Sometimes the whole family together is overwhelming. It's not that I don't love all of you. But sometimes an outing is best for a subset of people." I think the newness will wear off eventually and things will settle down. You might also try talking to your mom and asking her why the parade is so important to her and maybe getting a little better understanding of each other's POV.
Some of my family cannot fathom not being included in everything. But I'm OK with my sister having Thanksgiving with her ILs. Why shouldn't she? They're her family, too.
And you can say that, "You know, when you and SIL go out and don't include me, I know that's not personal, either. So please don't take it personally the other way around."