We present a complete classification of simply-connected pluriclosed manifolds with parallel Bismut torsion. Consequently, we also establish a splitting theorem for compact manifolds that are both pluriclosed with parallel Bismut torsion and have vanishing Bismut Ricci form. Due to the relations of these conditions with the Vaisman geometry, we also analyze the behavior of the pluriclosed flow, proving that it preserves the Vaisman condition on compact complex surfaces if and only if the starting metric has constant scalar curvature.