  We provide a characterization of parallelizable compact complex manifolds and
their quotients using holomorphic symmetric differentials. In particular we
show that compact complex manifolds of Kodaira dimension 0 having strongly
semiample cotangent bundle are parallelizable manifolds, while compact complex
manifolds of Kodaira dimension 0 having weakly semiample cotangent bundle are
quotients of parallelizable manifolds. The main constructions used involve
considerations about semiampleness of vector bundles, which are themselves of
interest. As a byproduct we prove that compact manifolds having Kodaira
dimension 0 and weakly sermiample cotangent bundle have infinite fundamental
group, and we conjecture that this should be the case for all compact complex
manifolds not of general type with weakly semiample cotangent bundle.
