In the theory of classical modular forms on the upper half plane, it is known how to compute the dimensions of the spaces of cusp forms of weight $k$ and level $\Gamma_0(N)$ for all integers $k, N$. For Siegel modular forms the dimensions in general are unknown, with some exceptions if the level is squarefree. Recently, Roy, Schmidt, and Yi were able to compute these dimensions in the case of Klingen congruence subgroups of level 4. A major component of the argument requires determining the "local" dimensions (for $p=2$) of spaces of fixed vectors for Klingen congruence subgroups of level $p^2$ in irreducible representations of $\text{GSp}(4)$ over the $p$-adic numbers, which had been done by Yi in their thesis. After discussing this, we will explain the computation of the associated local dimensions for Siegel congruence subgroups of level $p^2$, and contrast some of the phenomena that arise with the Klingen case.