Rotate the parallelepiped so that you are staring down its longest diagonal, or any diagonal really. Now the 3 faces of the parallelepiped directly facing you c

Rotate the parallelepiped so that you are staring down its longest diagonal, or any diagonal really. Now the 3 faces of the parallelepiped directly facing you can be cut into 6 congruent triangles. Slice down these 6 triangles to get 6 tetrahedra. it is then easy to show by Cavalieri's principle that that the volume of any of these tets is equal to the original tet. so the original tet has volume 1/6 of the parallepiped. However these 6 tets are not necessarily congruent, when I commented I was visualizing the case when the parallelepiped has equal side lengths, but in the general case the 6 tetrahedra are not congruent. However each of the 6 is congruent to at least 1 other out of the 6, since the parallepiped has reflection symmetry.