Variation of the string-theoretic mirror map q-series in families motivates an investigation of the Picard-Fuchs differential equation of a pencil under deformation. Such variations define special algebraic solutions to isomonodromic deformation equations. Explicit Painleve VI solutions coming from elliptic pencils are determined. The method is generalized, yielding a topological characterization of a large class of "pullback" solutions coming from specially parametrized Hurwitz curves.