  We study a general form of a degenerate or singular parabolic equation $$
u_t-|Du|^{\gamma}\big(\Delta u+(p-2)\Delta_\infty^Nu\big)=0 $$ that generalizes
both the standard parabolic $p$-Laplace equation and the normalized version
that arises from stochastic game theory. We develop a systematic approach to
study second order Sobolev regularity and show that $D^2u$ exists as a function
and belongs to $L^2_{\text loc}$ for a certain range of parameters. In this
approach proving the estimate boils down to verifying that a certain
coefficient matrix is positive definite. As a corollary we obtain, under
suitable assumptions, that a viscosity solution has a Sobolev time derivative
belonging to $L^2_{\text loc}$.
