  After investigating the $3$-dimensional case [35], we continue to address and
close the problems of optimal ball and horoball packings in truncated Coxeter
orthoschemes with parallel faces that exist in $n$-dimensional hyperbolic space
$\overline{\mathbb{H}}^n$ up to $n=6$. In this paper, we determine the optimal
ball and horoball packing configurations and their densities for the
aforementioned tilings in dimensions $4 \leq n \leq 6$, where the symmetries of
the packings are derived from the considered Coxeter orthoscheme groups.
Moreover, for each optimal horoball packing, we determine the parameter related
to the corresponding Busemann function, which provides an isometry-invariant
description of different optimal horoball packing configurations.
