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We prove interior H\"older estimate for the spatial gradients of theviscosity solutions to the singular or degenerate parabolic equation $$u_t=|\nabla u|^{\kappa}\mbox{div} (|\nabla u|^{p-2}\nabla u), $$ where $p\in(1,\infty)$ and $\kappa\in (1-p,\infty).$ This includes the from $L^\infty$ to$C^{1,\alpha}$ regularity for parabolic $p$-Laplacian equations in bothdivergence form with $\kappa=0$, and non-divergence form with $\kappa=2-p$.This work is a continuation of a paper by the last two authors \cite{JS}.

H\"older gradient estimates for a class of singular or degenerate parabolic equations