  In this paper, we obtain the interior derivative estimates of solutions for
elliptic and parabolic Hessian quotient equations. Then we establish the
Bernstein theorem for parabolic Hessian quotient equations, that is, any
parabolically convex solution $u=u(x,t)\in C^{4,2}(\mathbb{R}^n\times
(-\infty,0])$ for $-u_t\frac{S_n(D^2u)}{S_l(D^2u)}=1$ in $\mathbb{R}^n\times
(-\infty,0]$ must be the form of $u=-mt+P(x)$ with $m>0$ being a constant and
$P$ being a convex quadratic polynomial.
