I have never been suspended from an SE site so I don't know, but let's say that I was suspended from X.SE, and that was the parent site for my chat profile, on my chat profile it would then say
This user is suspended on the parent site and cannot chat for xxx days
However, this suggests that if I were to then change my parent site for my chat profile that I would be able to chat again if my new parent site hadn't suspended me.
Is that so? Or would I still not be able to chat? And what about if my X.SE profile was suspended, would my chat profile also be even if I had set the parent site there as Y.SE?
Also, if changing their parent site wouldn't make a difference then the text about them not being able to chat because they have been suspended on their parent site is a bit misleading as it suggests that changing their parent site would enable them to chat again.