  In this paper, we study the period mappings for the families of $K3$ surfaces
derived from the $3$-dimensional $5$-verticed reflexive polytopes. We determine
the lattice structures, the period differential equations and the projective
monodromy groups.
  Moreover, we show that one of our period differential equations coincides
with the unifomizing differential equation of the Hilbert modular orbifold for
the field $\mathbb{Q}(\sqrt{5})$.
