We introduce a class of parameter-dependent q-series arising from the moduli of certain pencils of elliptic curves and rank 19 lattice polarized K3 surfaces. These series serve to ‘link’ the various Hauptmoduln which arise as modular mirror maps (as characterized in [C. F. Doran, in The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), CRM Proc. Lect. Notes 24, 257–281 (2000; Zbl 0972.14033); Commun. Math. Phys. 212, 625–647 (2000; Zbl 1047.11043)]), and can in principle be computed from the functional invariants of the pencils. Each corresponds to an algebraic solution to an isomonodromic deformation equation which parametrizes the Hurwitz space for those invariants. The discussion is made more concrete in the context of the Painlevé VI equation.