  Assume that E and B are complex manifolds and that pi is a holomorphic Serre
fibration from E onto E such that E admits a finite dominating family of
holomorphic fiber-sprays over a small neighborhood of any point in B. We show
that the parametric Oka property (POP) of B implies POP of E; conversely, POP
of E implies POP of B for contractible parameter spaces. This follows from a
parametric Oka principle for holomorphic liftings which we establish in the
paper.
