  In this paper, by providing the uniform gradient estimates for a sequence of
the approximating equations, we prove the existence, uniqueness and regularity
of the conical parabolic complex Monge-Amp\`ere equation with weak initial
data. As an application, we prove a regularity estimates, that is, any
$L^{\infty}$-solution of the conical complex Monge-Amp\`ere equation admits the
$C^{2,\alpha,\beta}$-regularity.
