We study differential equations satisfied by modular forms
of two variables associated to $\Gamma_1\times \Gamma_2$, where
$\Gamma_i$ ($i=1,2$) are genus zero subgroups of $SL_2(\R)$ commensurable
with $SL_2(\Z)$, e.g., $\Gamma_0(N)$ or $\Gamma_0(N)^*$ for some $N$.
In some examples, these differential equations are realized as the
Picard-Fuchs differential equations of families of $K3$ surfaces with
large Picard numbers, e.g., $19, 18, 17, 16$. Our method rediscovers
some of the Lian-Yau examples of "modular relations" involving power
series solutions to the second and the third order differential equations
of Fuchsian type in \cite{LY1}, \cite{LY2}.