  Residual torsion-free nilpotence has proven to be an important property for
knot groups with applications to bi-orderability and ribbon concordance.
Mayland proposed a strategy to show that a two-bridge knot group has a
commutator subgroup which is a union of an ascending chain of parafree groups.
This paper proves Mayland's assertion and expands the result to the subgroups
of two-bridge link groups that correspond to the kernels of maps to
$\mathbb{Z}$. We call these kernels the Alexander subgroups of the links. As a
result, we show the bi-orderability of a large family of two-bridge link
groups. This proof makes use of a modified version of a graph theoretic
construction of Hirasawa and Murasugi in order to understand the structure of
the Alexander subgroup for a two-bridge link group.
