  We consider Pogorelov type estimates and Liouville type theorems to parabolic
$k$-Hessian equations of the form $-u_t \sigma_k (D^2u) =1$ in
$\mathbb{R}^n\times (-\infty, 0]$. We derive that any
\textbf{$k+1$-convex-monotone} solution to $-u_t \sigma_k (D^2u) =1$ when
$u(x,0)$ satisfies a quadratic growth and $0<m_1\le -u_t\le m_2$ must be a
linear function of $t$ plus a quadratic polynomial of $x$.
