  In this paper, we show that the constant-dimensional Weisfeiler-Leman
algorithm for groups (Brachter & Schweitzer, LICS 2020) can be fruitfully used
to improve parallel complexity upper bounds on isomorphism testing for several
families of groups. In particular, we show:
  - Groups with an Abelian normal Hall subgroup whose complement is
$O(1)$-generated are identified by constant-dimensional Weisfeiler-Leman using
only a constant number of rounds. This places isomorphism testing for this
family of groups into $\textsf{L}$; the previous upper bound for isomorphism
testing was $\textsf{P}$ (Qiao, Sarma, & Tang, STACS 2011).
  - We use the individualize-and-refine paradigm to obtain a
$\textsf{quasiSAC}^{1}$ isomorphism test for groups without Abelian normal
subgroups, previously only known to be in $\textsf{P}$ (Babai, Codenotti, &
Qiao, ICALP 2012).
  - We extend a result of Brachter & Schweitzer (arXiv, 2021) on direct
products of groups to the parallel setting. Namely, we also show that
Weisfeiler-Leman can identify direct products in parallel, provided it can
identify each of the indecomposable direct factors in parallel. They previously
showed the analogous result for $\textsf{P}$.
  We finally consider the count-free Weisfeiler-Leman algorithm, where we show
that count-free WL is unable to even distinguish Abelian groups in
polynomial-time. Nonetheless, we use count-free WL in tandem with bounded
non-determinism and limited counting to obtain a new upper bound of
$\beta_{1}\textsf{MAC}^{0}(\textsf{FOLL})$ for isomorphism testing of Abelian
groups. This improves upon the previous $\textsf{TC}^{0}(\textsf{FOLL})$ upper
bound due to Chattopadhyay, Tor\'an, & Wagner (ACM Trans. Comput. Theory,
2013).
