  Recent experiments observed a phase transition within the superconducting
regime of the heavy-fermion system CeRh$_2$As$_2$ when subjected to a $c$-axis
magnetic field. This phase transition has been interpreted as a parity
switching from even to odd parity as the field is increased, and is believed to
be of first order. If correct, this scenario provides a unique opportunity to
study the phenomenon of local nucleation around inhomogeneities in a
superconducting context. Here, we study such nucleation in the form of sharp
domain walls emerging on a background of spatially varying material properties
and hence, critical magnetic field. To this end, we construct a spatially
inhomogeneous Ginzburg-Landau functional and apply numerical minimization to
demonstrate the existence of localized domain wall solutions and study their
physical properties. Furthermore, we propose ultrasound attenuation as an
experimental bulk probe of domain wall physics in the system. In particular, we
predict the appearance of an absorption peak due to domain wall percolation
upon tuning the magnetic field across the first-order transition line. We argue
that the temperature dependence of this peak could help identify the nature of
the phase transition.
