  In this paper we establish some parabolicity criteria for maximal surfaces
immersed into a Lorentzian product space of the form $M^2\times\mathbb{R}_1$,
where $M^2$ is a connected Riemannian surface with non-negative Gaussian
curvature and $M^2\times\mathbb{R}_1$ is endowed with the Lorentzian product
metric $<,>=<,>_M-dt^2$. In particular, and as an application of our main
result, we deduce that every maximal graph over a starlike domain
$\Omega\subseteq M$ is parabolic. This allows us to give an alternative proof
of the non-parametric version of the Calabi-Bernstein result for entire maximal
graphs in $M^2\times\mathbb{R}_1$.
