On balanced Hermitian threefolds with parallel Bismut torsion

We continue our study on Hermitian manifolds that are {\em Bismut torsion parallel,} or {\em BTP} for brevity, which means that the Bismut connection has parallel torsion tensor. For $n\ge 3$, BTP metrics can be balanced (and non-Kähler). In this paper, we give a detailed description to characterize all compact, balanced BTP threefolds.