  Extending a recent theory developed on compact K\"ahler manifolds by
Guedj-Lu-Zeriahi and the author, we define and study pluripotential solutions
to degenerate parabolic complex Monge-Amp\`ere equations on compact Hermitian
manifolds. Under natural assumptions on the Cauchy boundary data, we show that
the pluripotential solution is semi-concave in time and continuous in space and
that such a solution is unique. We also establish a partial regularity of such
solutions under some extra assumptions of the densities and apply it to prove
the existence and uniqueness of the weak Chern-Ricci flow on complex compact
varieties with log terminal singularities.
