  We introduce a new class of quasi-linear parabolic equations involving
nonhomogeneous degeneracy or/and singularity $$ \partial_t u=[|D u|^q+a(x,t)|D
u|^s]\left(\Delta u+(p-2)\left\langle D^2 u\frac{D u}{|D u|},\frac{D u}{|D
u|}\right\rangle\right), $$ where $1<p<\infty$, $-1<q\leq s<\infty$ and
$a(x,t)\ge 0$. The motivation to investigate this model stems not only from the
connections to tug-of-war like stochastic games with noise, but also from the
non-standard growth problems of double phase type. According to different
values of $q,s$, such equations include nonhomogeneous degeneracy or
singularity, and may involve these two features simultaneously. In particular,
when $q=p-2$ and $q<s$, it will encompass the parabolic $p$-Laplacian both in
divergence form and in non-divergence form. We aim to explore the from
$L^\infty$ to $C^{1,\alpha}$ regularity theory for the aforementioned problem.
To be precise, under some proper assumptions, we use geometrical methods to
establish the local H\"{o}lder regularity of spatial gradients of viscosity
solutions.
