  In this paper, we establish the gradient and Pogorelov estimates for
$k$-convex-monotone solutions to parabolic $k$-Hessian equations of the form
$-u_t\sigma_k(\lambda(D^2u))=\psi(x,t,u)$. We also apply such estimates to
obtain a Liouville type result, which states that any $k$-convex-monotone and
$C^{4,2}$ solution $u$ to $-u_t\sigma_k(\lambda(D^2u))=1$ in
$\mathbb{R}^n\times(-\infty,0]$ must be a linear function of $t$ plus a
quadratic polynomial of $x$, under some growth assumptions on $u$.
