  We consider the (discrete) parabolic Anderson model $\partial u(t,x)/\partial
t=\Delta u(t,x) +\xi_t(x) u(t,x)$, $t\geq 0$, $x\in \mathbb{Z}^d$. Here, the
$\xi$-field is $\mathbb{R}$-valued, acting as a dynamic random environment, and
$\Delta$ represents the discrete Laplacian. We focus on the case where $\xi$ is
given by a rescaled symmetric simple exclusion process which converges to an
Ornstein--Uhlenbeck process. By scaling the Laplacian diffusively and
considering the equation on a torus, we demonstrate that in dimension $d=2$,
when a suitably renormalized version of the above equation is considered, the
sequence of solutions converges in law. This resolves an open problem
from~\cite{EH23}, where a similar result was shown in the three-dimensional
case. The novel contribution in the present work is the establishment of a
gradient bound on the transition probability of a fixed but arbitrary number of
labelled exclusion particles.
