Perfect parallelograms have edge lengths and diagonal lengths that are all positive integers. These generalize Pythagorean triples which are perfect rectangles. We consider the distribution of perfect parallelograms and show they satisfy a quadratic Diophantine equation. The solutions to that Diophantine equation can be generated by a finite collection of matrices that generalizes the matrix based tree of Pythagorean triples.

perfect parallelepiped, perfect cuboid, Barning tree, generalized Pythagorean triangles.