  The exact complexity of solving parity games is a major open problem. Several
authors have searched for efficient algorithms over specific classes of graphs.
In particular, Obdr\v{z}\'{a}lek showed that for graphs of bounded tree-width
or clique-width, the problem is in $\mathrm{P}$, which was later improved by
Ganardi, who showed that it is even in $\mathrm{LOGCFL}$ (with an additional
assumption for clique-width case). Here we extend this line of research by
showing that for graphs of bounded tree-depth the problem of solving parity
games is in logspace uniform $\text{AC}^0$. We achieve this by first
considering a parameter that we obtain from a modification of clique-width,
which we call shallow clique-width. We subsequently provide a suitable
reduction.
