  We completely characterize the covers of connected orbifold curves which
preserve slope stability of vector bundles under the pullback morphism. More
precisely, given a cover $f \colon (Y,Q) \longrightarrow (X,P)$ of connected
orbifold curves, we show that the maximal destabilizing sub-bundle of the
pushforward sheaf $f_*\mathcal{O}_{(Y,Q)}$ defines the maximal \'{e}tale
sub-cover of $f$. The cover $f$ is said to be genuinely ramified if $f$ does
not factor through any non-trivial \'{e}tale sub-cover. Our main result states
that the class of covers $f$ that preserves the stable bundles under a pullback
are precisely the class of genuinely ramified covers $f$. Further, we establish
equivalent conditions for the cover $f$ to be genuinely ramified, generalizing
earlier works on covers of curves. We thoroughly study the slope stability
conditions of bundles on an orbifold curve, their properties under the
pushforward and pullback maps under covers with a stand point of
Deligne-Mumford stacks, hence giving a solid foundation of the subject. As a
consequence, we also answer the question of descent of stable bundles under
genuinely ramified covers.
