Cheng-Yau proved that: A complete Riemannian manifold with non-negative Ricci curvature and at most quadratic growth for volumes of balls as the radius goes to infinity is parabolic.
EDIT (by Igor Belegradek). Various criteria for parabolicity are found in the survey of Grigoryan. In particular, on page 177 it is mentioned that a a complete Riemannian manifold with at most quadratic volume growth is parabolic. For complete open nonnegatively curved surfaces the volume growth is at most quadratic by Bishop-Gromov. On the other hand, there are parabolic complete manifolds with arbitrary fast volume growth (see page 180 of the same survey). Finally, the very first proof of parabolicity of complete nonnegatively curved plane seems to be due to Blanc-Fiala (1941); the reference is in Huber's paper mentioned in my comment to Anton's answer.