Your video is a bit fast to follow. It looks like at the point that you vertex parent your smaller cube to your bigger cube, your smaller cube jumps? This can happen in a few situations, but usually when an object has been parented to something else previously. It's a bug, but a small, easily rectified one: undo the parent operation, then clear parent inverse on the small cube (alt p -> clear parent inverse) then parent it to the big cube again.
It moves with the cube's rotation, but it doesn't rotate with the cube's rotation, because it is parented to a vertex, and rotation just isn't a property of something 0-dimensional like a point. Blender can't tell you or anyone what way that vertex parent is rotated. If you want the small cube to acquire rotation from the vertex parent, you need to parent it to three vertices. Select the big cube, enter edit, select three and only three vertices on it, then in object mode, parent the small cube to the big cube with vertex (triangle) option.