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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 alinear function of $t$ plus a quadratic polynomial of $x$.

A Pogorelov estimate and a Liouville type theorem to parabolic $k$-Hessian equations