Area of parallelogram = Area of square. Shear transform

Behold, $\phantom{proof without words}$

Slice each figure by infinitely many infinitely thin horizontal layers. The area of each slice is the same as that of the corresponding slice in the original square: corresponding slices both have the same width and height and the ends can be neglected (in the given limit).

Just for completeness, here is another method of disection that proves the result.