Surface imperfections and play in the hinge aside, this would not work as you might imagine. This configuration (as illustrated), on a flat surface, would only completely image the zenith. Put it on a mount and it would image whatever is directly above the flat mirror. As soon as you move the hinged mirror, those light rays that eventually would lead to the camera sensor will only go flying into the tube walls. The light doesn't magically find its way to the parabolic mirror at any hinge position. A couple degrees around 45°, you would probably still get an image, but only at a 45° incline on the hinge would you get a complete image.
Surface imperfections and play in the hinge aside, this would not work as you might imagine. This configuration (as illustrated), on a flat surface, would only completely image the zenith. Put it on a mount and it would image whatever is directly above the flat mirror. As soon as you move the hinged mirror, those light rays that eventually would lead to the camera sensor will only go flying into the tube walls. The light doesn't magically find its way to the parabolic mirror at any hinge position. A couple degrees around 45°, you would probably still get an image, but only at a 45° incline on the hinge would you get a complete image.