it would be better if you could give an illustration..
This result is more or less obvious if the ellipse happens to be a circle. The circumscribing parallelogram will be a rhombus, and the second parallelogram will be a rectangle formed by joining the points of contact of the rhombus with the circle.
But you can get any ellipse from a circle by a contraction in the direction of the minor axis. Such a contraction takes straight lines to straight lines and preserves parallelism (though it does not preserve lengths or angles). So it will take the circumscribing rhombus to a parallelogram, and the inner rectangle to a parallelogram whose sides are parallel to the diagonals of the outer parallelogram. Any circumscribing parallelogram arises in this way.