  In this article, we study Hermitian manifolds whose Bismut-Strominger
connection has parallel torsion tensor, which will be called {\em Bismut
torsion parallel manifolds,} or {\em BTP} manifolds for short. We obtain a
necessary and sufficient condition characterizing this class in terms of the
curvature tensor. In particular, Bismut flat or Bismut K\"ahler-like manifolds
are {\em BTP} manifolds, known by our earlier results. We also obtain
structural results for non-balanced {\em BTP} manifolds, and classification
theorems for non-balanced {\em BTP} threefolds and balanced {\em BTP}
threefolds.
