It's called reciprocity, and it's really cool!
We decided to go with the Western North Carolina Nature Center membership, because it was affordable and offered the best balance we could find for what we were looking for.
List of Participating AZA Zoos and Aquarium by state, and their benefits (look for ones that have 100% & 50%. That means you get in with a family of 4 at no cost).
Map to find the nearest AZA Participating facility near your location
This map is super handy to find the zoos and aquariums that participate [Added 4/18/2019]