A weighted manifold is a Riemannian manifold $M^m$ together with a $C^1$ function $e^h$, which is used to weight the Hausdorff measures associated to the Riemannian metric. The $h$-parabolicity condition in a weighted manifold is the Liouville-type property that any function $u$ bounded from above and $h$-subharmonic, i.e., $\Delta^hu\geq0$, must be constant. Here, $\Delta^hu=\Delta u+\langle\nabla h,\nabla u\rangle$ is the weighted Laplacian operator. If the $h$-parabolicity condition fails, then $M$ is said to be $h$-hyperbolic. In this paper, the authors consider weights in rotationally symmetric manifolds with a pole (that they call model spaces) and establish some criteria ensuring $h$-parabolicity or $h$-hyperbolicity for submanifolds of model spaces under restrictions on the geometry of the model, the ambient weight, and the weighted mean curvature of the submanifold. Their results are obtained from capacity comparisons with respect to weighted model spaces, which are model spaces together with a radial weight. \par Quite fascinatingly, the authors recover an Ahlfors-type result, already shown by A.~A. Grigorʹyan [in {\it The ubiquitous heat kernel}, 93--191, Contemp. Math., 398, Amer. Math. Soc., Providence, RI, 2006; MR2218016], which describes the parabolicity of a weighted model space by means of an integrability condition for the weighted area function of the metric spheres centered at the pole. They also prove geometric restrictions and characterization theorems for submanifolds with controlled weighted mean curvature, generalizing some half-space and Bernstein-type theorems. Furthermore, they show a direct application of their parabolicity criteria to the classification of stable hypersurfaces in a weighted sense.