If a parallelogram is circumscribing an ellipse, the point of contacts are the vertices of a parallelogram. How can one prove that the diagonals of the first parallelogram are parallel to the sides of the second parallelogram?
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.