I believe I know the issue. Geo parented to camera does move when the camera is moved. Look at my gif below. As I transform the camera the geometry moves with it. The issue has to do with locking the camera to the viewport. Notice if you use the view tool the camera's transform won't get updated until mouse release. You can fix that by enabling "export view continuously". The docs say
By default, the view is exported to the camera or light when the mouse button is released. This behavior can be overridden by turning on Export View Continuously in the camera menu. This option should be used with caution as exporting the view can trigger cooking.
Hope this helps