Let \((M,g)\) be a compact Hermitian manifold of complex dimension \(n\) with \(\text{Vol}(M) = 1\), and \(F\) a smooth function on \(M\). The main result of this article shows that there is a smooth solution \(\varphi\) to the parabolic complex Monge-Ampère equation
\[ \frac{\partial \varphi}{\partial t} = \log\frac{\det(g^{}_{i\bar j} + \partial^{}_i \partial^{}_{\bar j} \varphi)}{\det g^{}_{i\bar j}} - F, \quad g^{}_{i\bar j} + \partial^{}_i \partial^{}_{\bar j} \varphi>0, \]
for all time. Furthermore, \(\widetilde{\varphi}\) converges in \(C^\infty\) to a smooth function \(\widetilde{\varphi}_\infty\), where \[ \widetilde{\varphi} = \varphi - \int_M \varphi \omega^n. \] The author develops an argument showing that there is a unique real number \(b\) such that the pair \((b, \widetilde{\varphi}_\infty)\) is the unique solution to the Monge-Ampère equation
\[ \log\frac{\det(g^{}_{i\bar j} + \partial^{}_i \partial^{}_{\bar j} \varphi)}{\det g^{}_{i\bar j}} = F + b, \quad g^{}_{i\bar j} + \partial^{}_i \partial^{}_{\bar j} \varphi>0, \]
with \(\int_M \varphi \omega^n = 0\). This is an extension to a result by P. Cherrier [Bull. Sci. Math., II. Sér. 111, 343–385 (1987; Zbl 0629.58028)].