  The minimal coloring number of a $\mathbb{Z}$-colorable link is the minimal
number of colors for non-trivial $\mathbb{Z}$-colorings on diagrams of the
link. In this paper, we show that the minimal coloring number of any
non-splittable $\mathbb{Z}$-colorable links is four. As an example, we consider
the link obtained by replacing each component of the given link with several
parallel strands, which we call a parallel of a link. We show that an even
parallel of a link is $\mathbb{Z}$-colorable except for the case of 2 parallels
with non-zero linking number. We then give a simple way to obtain a diagram
which attains the minimal coloring number for such even parallels of links.
