You must have $h^{n0}=1$, because you have a parallel volume form, so nowhere zero, and any other holomorphic volume form is a holomorphic multiple, so a constant multiple (as holomorphic functions are constant).
Kobayashi, First Chern class and holomorphic tensor fields, 1980, proved that any holomorphic tensor on a compact Ricci-flat Kaehler manifold is parallel. Hence it induces a reduction of holonomy group. The holonomy group is a product of groups arising as irreducible holonomy groups, i.e. copies of the special unitary group $SU(n_i)$ and the compact symplectic group $Sp(n_j)$. But since you have $h^{20}=0$, you have no $Sp(n_j)$ factors, so a product of $SU(n_i)$ factors, i.e. locally a product of Calabi-Yau and flat Kaeher metrics. Unless there is a flat factor in the product, you don't get any parallel tensors except for the holomorphic $n_j$-forms. By a theorem of Beauville (J.D.G. 1983), after a finite covering, you have a product of simply connected Calabi Yaus with irreducible $SU(n_i)$ holonomy, $n_i\ge 3$, and a flat complex torus. But to have $h^{20}=0$, that torus is an elliptic curve.