  Given a discrete subgroup $\Gamma$ of finite co-volume of
$\mathrm{PGL}(2,\mathbb{R})$, we define and study parabolic vector bundles on
the quotient $\Sigma$ of the (extended) hyperbolic plane by $\Gamma$. If
$\Gamma$ contains an orientation-reversing isometry, then the above is
equivalent to studying real and quaternionic parabolic vector bundles on the
orientation cover of $\Sigma$. We then prove that isomorphism classes of
polystable real and quaternionic parabolic vector bundles are in bijective
correspondence with equivalence classes of real and quaternionic unitary
representations of $\Gamma$. Similar results are obtained for compact-type real
parabolic vector bundles over Klein surfaces.
