Hi,
In my experience these kind of issues originate from bad mesh quality, i.e. collapsed elements.
The parabolic system is more resistant, as it is diagonal dominant (and even more so with mass lumping), but then solvability problems pop up when changing parallel layout or region parameters. One thinks that the issue was introduced by the change in parameters, while it has been present all the time.
To test my hypothesis, I would suggest running "meshtool query quality" on your input mesh, to sanity check it. The max. quality metric should not be over 0.98.
Best wishes, Aurel