  Our aim in this article is to produce new examples of semistable Lazarsfeld-
Mukai bundles on smooth projective surfaces $X$ using the notion of parabolic
vector bundles. In particular, we associate natural parabolic structures to any
rank two (dual) Lazarsfeld-Mukai bundle and study the parabolic stability of
these parabolic bundles. We also show that the orbifold bundles on Kawamata
coverings of $X$ corresponding to the above parabolic bundles are themselves
certain (dual) Lazarsfeld-Mukai bundles. This gives semistable Lazarsfeld-Mukai
bundles on Kawamata covers of the projective plane and of certain K3 surfaces.
