We consider viscosity solutions to nonlinear uniformly parabolic equations in nondivergence form on a Riemannian manifold $M$ with the sectional curvature bounded from below by $−κ$ for $κ≥0.$ In the elliptic case, Wang and Zhang [24] recently extended the results of [5] to nonlinear elliptic equations in nondivergence form on such $M$, where they obtained the Harnack inequality for classical solutions. We establish the Harnack inequality for nonnegative viscosity solutions to nonlinear uniformly parabolic equations in nondivergence form on $M$. The Harnack inequality of nonnegative viscosity solutions to the elliptic equations is also proved.