We study mirror symmetry of families of elliptic K3 surfaces with elliptic fibers of type E6, E7 and E8. We consider a moduli space T of the mirror K3 surfaces enhanced with the choice of differential forms. We show that coordinates on T are given by the ring of quasi modular forms in two variables, with modular groups adapted to the fiber type. We furthermore introduce an algebraic group G which acts on T from the right and construct its Lie algebra Lie(G). We prove that the extended Lie algebra generated by Lie(G) together with modular vector fields on T is isomorphic to sl2(C) sl2(C).