  We consider energy solutions of the inhomogeneous parabolic $p$-Laplacien
system $\partial_t u-\text{div}(|D u|^{p-2}D u)=-\text{div} g$. We show in the
case $p\geq 2$ that if the right hand side $g$ is locally in
$L^\infty(\text{BMO})$, then $u$ is locally in $L^\infty(\mathcal{C}^1)$, where
$\mathcal{C}^1$ is the 1-H\"older--Zygmund space. This is the borderline case
of the Calder\'on-Zygmund theorey. We provide local quantitative estimates. We
also show that finer properties of $g$ are conserved by $D u$, e.g.\ H\"older
continuity.
  Moreover, we prove a new decay for gradients of $p$-caloric solutions for all
$\frac{2n}{n+2}<p<\infty$.
