  We construct projectivization of a parabolic vector bundle and a tautological
line bundle over it. It is shown that a parabolic vector bundle is ample if and
only if the tautological line bundle is ample. This allows us to generalize the
notion of a k-ample bundle, introduced by Sommese, to the context of parabolic
bundles. A parabolic vector bundle $E_*$ is defined to be k-ample if the
tautological line bundle ${\mathcal O}_{{\mathbb P}(E_*)}(1)$ is $k$--ample. We
establish some properties of parabolic k-ample bundles.
