Parabolic Lie algebroid connections on parabolic principal bundles over curves

Let $X$ be a compact connected Riemann surface and $S\,\subset\, X$ a finite subset. We consider parabolic principal $G$--bundles $\mathcal{E}_{G}$ on $X$ with parabolic structure on $S$, where $G$ is a connected complex reductive affine algebraic group. Let $P\, \subset\, G$ be a parabolic subgroup and $\mathcal{E}_{P}\, \subset\, \mathcal{E}_{G}$ a reduction of structure group of $\mathcal{E}_{G}$ to $P$. We give a criterion for the existence of a parabolic Lie algebroid connection on $\mathcal{E}_{P}$ for any given parabolic Lie algebroid on $(X,\,S)$ whose anchor map is not surjective. More precisely, $\mathcal{E}_{P}$ admits a parabolic Lie algebroid connection if the reduction $\mathcal{E}_{P}\, \subset\, \mathcal{E}_{G}$ is parabolically infinitesimally rigid. In particular, the Harder--Narasimhan reduction of $\mathcal{E}_{G}$ admits a parabolic Lie algebroid connection.