Reference for Dyck languages being $\mathsf{TC}_0$-complete
Here is an $AC_0$ reduction from $\rm Majority$ to $Dyck(1)$.
(This implies that $\rm Majority$ is $AC_0$ reducible to $Dyck(k)$ for all $k \geq 1$.)
In order to do it, we construct a poly-size
constant depth circuit whose gates are $AND$, $OR$, $NOT$ and $Dyck(1)$.




Given an instance $x \in \{0,1\}^n$ of $\rm Majority$ do
Compute $y \in \{0,1\}^{2n}$ by replacing each $0$ with $(($ and each $1$ with $()$.
Now for each $i = 1,\dots, n/2$ let $z_i$ be the string obtained by concatenating $y$ with $2i$-many closing parentheses, i.e. $z_i = y \circ )^{2i}$.
If $z_i \in Dyck(1)$ for some $i = 1,\dots, n/2$ then ACCEPT.
Otherwise, REJECT.




This can be clearly done with a constant depth circuit. (Computing $z_i$ can be done in depth 1, and computing the last step is done using an $OR$ gate.)

It is also easy to see that this circuit indeed computes $\rm Majority$ because $z_i \in Dyck(1)$ if and only if ${\rm weight}(x) = n - i$.