I prefer the A.O. to the side parallax adjustment. I think the A.O. has a better depth of field
I didn't read all the posts but this is it for me. Depth of field. The side focus scopes I have owned were the newer Leupolds, Zeiss and Nikon.
To me the best side focus was the Nikon Buckmaster.......
I have no clue to all the technical reasons just I know what I saw....
Dave