Amicable Heronian Parallelograms

A convex polygon is Heronian if its side lengths and its area are integers. Two polygons are amicable if the area of one is equal to the perimeter of the other, and vice versa. We show that there are infinitely many pairs of amicable Heronian parallelograms, and we give necessary and sufficient conditions for a Heronian parallelogram to be part of an amicable pair.