Direct image and parabolic structure on symmetric product of curves

Given a complex smooth projective curve X and a vector bundle E on it; there is a corresponding vector bundle F(E) on the symmetric product Sn(X) for any n. We show that there is a natural parabolic structure on the vector bundle F(E). We prove that F(E) is parabolic semistable (respectively, parabolic polystable) if and only if E is semistable (respectively, polystable). If E is not the trivial line bundle on X, then we prove that F(E) is parabolic stable if and only if E is stable.