The opposite sides must be parallel for the intersection to be a parallelogram. This means that the opposite sides must lie on parallel faces of a cube. Let's consider the top and bottom faces of the cube. We can use an opposing vertex from each and the midpoint of an edge opposite to it to vertex. Then, consider a plane through containing these points.
Since the opposite sides lie on parallel faces of the cube, the sides are parallel. This means that the cross section is a parallelogram. Moreover, the interior angles of the cross section are not perpendicular, so the cross section is not a rectangle. As we can see, it is possible to obtain a non-rectangular parallelogram as a cross section from a cube.
