Other correct solutions were sent in by Andrew Hess, Carl Westine, Greg Rutter. Further solutions were submitted by Burkart Venzke, Aaron David Kahn, Andrew Kapust, Ivan Lisac, Tim Kelley.

Starting with the yellow parallelogram having integral vertices, draw in the remaining triangles and squares as shown.  The area of the resulting rectangle is an integer as is the area of each of the new pieces.  Therefore the area of the original must be an integer, too.
 

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