  In \cite{BR1}, \cite{BR2}, a parabolic determinant line bundle on a moduli
space of stable parabolic bundles was constructed, along with a Hermitian
structure on it. The construction of the Hermitian structure was indirect: The
parabolic determinant line bundle was identified with the pullback of the
determinant line bundle on a moduli space of usual vector bundles over a
covering curve. The Hermitian structure on the parabolic determinant bundle was
taken to be the pullback of the Quillen metric on the determinant line bundle
on the moduli space of usual vector bundles. Here a direct construction of the
Hermitian structure is given. For that we need to establish a version of the
correspondence between the stable parabolic bundles and the Hermitian--Einstein
connections in the context of conical metrics. Also, a recently obtained
parabolic analog of Faltings' criterion of semistability plays a crucial role.
