We discuss fully nonlinear second order uniformly parabolic equations including Isaacs parabolic equations. Isaacs equations arise in the theory of stochastic differential games. In 2014, N.V. Krylov proved the existence of $L^p$-viscosity solutions of boundary value problems for equations with VMO (vanishing mean oscillation) "coefficients" when $p>n+2$. Furthermore, the solutions were in the parabolic Hölder space $C^{1+\alpha, \frac{1+\alpha}{2}}$ for $\alpha\in(0, 1)$. Our purpose is to show interior Hölder estimates on the spatial gradients of $L^p$-viscosity solutions of fully nonlinear parabolic equations under the same conditions as in Krylov's result.