We give a correspondence between toric 3-Sasaki 7-manifolds S and certain toric Sasaki-Einstein 5-manifolds M. These 5-manifolds are all diffeomorphic to # k(S²× S³), where k=2b₂(S)+1, and are given by a pencil of Sasaki embeddings, where M⊂S is given concretely by the zero set of a component of the 3-Sasaki moment map. It follows that there are infinitely many examples of these toric Sasaki-Einstein manifolds M for each odd b₂(M)>1. This is proved by determining the invariant divisors of the twistor space Z of S, and showing that the irreducible such divisors admit orbifold Kähler-Einstein metrics.

As an application of the proof we determine the local space of anti-self-dual structures on a toric anti-self-dual Einstein orbifold.