  We study regularity results for nonlinear parabolic systems of $p$-Laplacian
type with inhomogeneous boundary and initial data, with
$p\in(\frac{2n}{n+2},\infty)$. We show bounds on the gradient of solutions in
the Lebesgue-spaces with arbitrary large integrability exponents and natural
dependences on the right hand side and the boundary data. In particular, we
provide a new proof of the global non-linear Calder\'on-Zygmund theory for such
systems. Our method makes use of direct estimates on the solution minus its
boundary values and hence is considerably shorter than the available higher
integrability results. Technically interesting is the fact that our parabolic
estimates have no scaling deficit with respect to the leading order term.
Moreover, in the singular case, $p\in(\frac{2n}{n+2},2]$, any scaling deficit
can be omitted.
