  In this paper, we study the implications of the modular $A^{}_4$ flavor
symmetry in constructing a supersymmetric minimal type-(I+II) seesaw model, in
which only one right-handed neutrino and two Higgs triplets are introduced to
account for the tiny neutrino masses, flavor mixing and CP violation. The
right-handed neutrino as well as the Higgs triplets in this model are assigned
into the trivial one-dimensional irreducible representation of the modular
group $A^{}_{4}$. We show that the individual contributions to the neutrino
masses from the right-handed neutrino and the Higgs triplet are comparable. We
also find that the neutrino mass matrix can possess an approximate $\mu-\tau$
reflection symmetry for some specific values of free model parameters.
Moreover, our model predicts relatively large masses of three light neutrinos,
thus can be easily tested in future neutrino experiments.
