  We obtain higher order estimates for a parabolic flow on a compact Hermitian
manifold. As an application, we prove that a bounded
$\hat{\omega}$-plurisubharmonic solution of an elliptic complex
Monge-Amp\`{e}re equation is smooth under an assumption on the background
Hermitian metric $\hat{\omega}$. This generalizes a result of Sz\'{e}kelyhidi
and Tosatti on K\"{a}hler manifolds.
