We extend the notion of lattice polarization for K3 surfaces to families in a way that gives control over the action of monodromy on the algebraic cycles, and discuss the uses of this new theory in the study of families of K3 surfaces admitting fiberwise symplectic automorphisms. We then give an application of these ideas to the study of Calabi-Yau threefolds admitting fibrations by lattice polarized K3 surfaces. This approach sheds new geometric light on a recent question of Sarnak about the dichotomy between arithmetic/thin monodromy representations for the 14 “mirror-compatible” variations of Hodge structure underlying families of Calabi-Yau threefolds over the thrice-punctured sphere classified in the work of Doran-Morgan.