Direct image of structure sheaf and parabolic stability

Let $f:X\to Y$ be a dominant generically smooth morphism between irreducible smooth projective curves over an algebraically closed field $k$ such that $\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{r}(k)>\text{degree}(f)$ if the characteristic of $k$ is nonzero. We prove that $(f_{\ast }{\mathcal{O}}_X)\mathrm{/}{\mathcal{O}}_Y$ equipped with a natural parabolic structure is parabolic polystable. Several conditions are given that ensure that the parabolic vector bundle $(f_{\ast }{\mathcal{O}}_X)\mathrm{/}{\mathcal{O}}_Y$ is actually parabolic stable.