I think "parking" is something that most sloop-rigged monohulls cannot do. At least they do not seem to do it. If you are at a regatta with monohull sloops (Flying Scots, for instance), you do not see them parking. They just keep sailing back and forth all the time between races.
Most monohull dinghies can indeed park. They teach how in learn-to-sail classes (US Sailing's included) and call it the "safety position".
That said, I too do not recall seeing monohulls park between races like cats do. I suspect it may be because (1) monohull sailors don't work as hard while racing and aren't so tired, and maybe (2) monohulls don't take off on you like cats do when you aren't paying attention.
I know the Tanzer 16 certainly doesn't. Then again, it's so slow that you simply don't have much time to wait between races. Maybe that's it.
