No idea about the Parrot gun, although it might be the name of the chap that invented it 
...
True enough, but, as I thought, it was that weird cylinder at the widest., a WIKI entry might clarify.
"The cast iron made for an accurate gun, but was brittle enough to suffer fractures. Hence, a large wrought iron reinforcing band was overlaid on the breech to give it additional strength."
Course, I also thought the barrel taper was a bit curved, but not finding that.
Doug