  We investigate the $C^{1+\alpha}$-regularity of solutions of parabolic
equations $\partial_{t}v+H(v,Dv,D^{2}v,t,x)=0$. Our main result says that under
rather general assumptions there exist $C$-viscosity and $L_{p}$-viscosity
solutions which are in $C^{1+\alpha}_{loc}$. We allow $H$ to be just measurable
in $t$ and for its principal part to have sufficiently small discontinuities as
a function of~$x$. No Lipschitz continuity of $H$ with respect to $v,Dv$ is
required.
