The easiest way to answer your question is by using the Brownian motion on the manifold. In these terms parabolicity of an end means its recurrence with respect to the Brownian motion. The reason why presence of at least two non-parabolic ends (not "more than two" as you write) implies existence of non-constant bounded harmonic function (boundedness condition which you don't mention is important - otherwise all these claims fail) is then very simple. The Brownian sample paths escape to infinity along one of non-parabolic ends, so that if there are at least two of them there will be a non-trivial behaviour at infinity, i.e., a non-constant bounded harmonic function.

On the other hand, if there is just one non-parabolic end, then the space of bounded harmonic functions on the whole manifold is the same as just for this end, so that in this case both situations are possible.