  In this paper, we prove gradient continuity estimates for viscosity solutions
to $\Delta_{p}^N u- u_t= f$ in terms of the scaling critical $L(n+2,1 )$ norm
of $f$, where $\Delta_{p}^N$ is the game theoretic normalized $p-$Laplacian
operator defined in (1.2) below. Our main result, Theorem 2.5 constitutes
borderline gradient continuity estimate for $u$ in terms of the modified
parabolic Riesz potential $\mathbf{P}^{f}_{n+1}$ as defined in (2.8) below.
Moreover, for $f \in L^{m}$ with $m>n+2$, we also obtain H\"older continuity of
the spatial gradient of the solution $u$, see Theorem 2.6 below. This improves
the gradient H\"older continuity result in [3] which considers bounded $f$. Our
main results Theorem 2.5 and Theorem 2.6 are parabolic analogues of those in
[9]. Moreover differently from that in [3], our approach is independent of the
Ishii-Lions method which is crucially used in [3] to obtain Lipschitz estimates
for homogeneous perturbed equations as an intermediate step.
