Please ignore this post, see solution below...
Wow, you were right. It's very, very bad that you were right
So it appears you cannot use a stationary (0/0 parallax) HUD layer for your controls.
So unless there is some trick documented in fine print, to use the system-pick-overlapping-at-point function, you cannot have any parallax at all. You will have to add code to the every tick event to reposition your HUD controls.