Orbifold principal bundles on an elliptic fibration and parabolic principal bundles on a Riemann surface

Let $X$ be a compact Riemann surface and associated to each point $p_i$ of a finite subset $S$ of $X$ is a positive integer $m_i$. Fix an elliptic curve $C$. To this data we associate a smooth elliptic surface $Z$ fibered over $X$. The group $C$ acts on $Z$ with $X$ as the quotient. It is shown that the space of all vector bundles over $Z$ equipped with a lift of the action of $C$ is in bijective correspondence with the space of all parabolic bundles over $X$ with parabolic structure over $S$ and the parabolic weights at any $p_i$ being integral multiples of $1/m_i$. A vector bundle $V$ over $Z$ equipped with an action of $C$ is semistable (respectively, polystable) if and only if the parabolic bundle on $X$ corresponding to $V$ is semistable (respectively, polystable). This bijective correspondence is extended to the context of principal bundles