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?
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.