  We introduce a notion of admissible Hermitian metrics on parabolic bundles
and define positivity properties for the same. We develop Chern-Weil theory for
parabolic bundles and prove that our metric notions coincide with the already
existing algebro-geometric versions of parabolic Chern classes. We also
formulate a Griffiths conjecture in the parabolic setting and prove some
results that provide evidence in its favour for certain kinds of parabolic
bundles. For these kinds of parabolic structures, we prove that the conjecture
holds on Riemann surfaces. We also prove that a Berndtsson-type result holds,
and that there are metrics on stable bundles over surfaces whose Schur forms
are positive.
