  In this work we establish new equivalences for the concept of $p$-parabolic
Riemannian manifolds. We define a concept of comparison principle for elliptic
PDE's on exterior domains of a complete Riemannian manifold $M$ and prove that
$M$ is $p$-parabolic if and only if this comparison principle holds for the
$p$-Laplace equation. We show also that the $p$-parabolicity of $M$ implies the
validity of this principle for more general elliptic PDS's and, in some cases,
these results can be extended for non $p$-parabolic manifolds or unbounded
solutions, provided that some growth of these solutions are assumed.
