Changing the Parent of a stringgrid from one frame to another successfully moves display of the stringgrid to the new frame here (Laz 1.4.0, win32).
However, you're planning on destroying the original frame, which remains the owner of the stringgrid, so at that point the grid will also be destroyed.
Presumably you could dynamically create the grid, and so make it owned by some other instance than the frame you are destroying. But if you use the RAD designer to create the grid by dropping it on a frame from the component palette, its owner will be set to the original frame.