Smallest polyhedral tilings of 3-tori by parallelohedra

There are two parallelohedra in dimension 2 and five in dimension 3. We study smallest polyhedral quotients of the tilings defined by them on tori. For 2-tori the situation is well understood, and the three minimal models are well known. Here we investigate the five 3-dimensional cases with the result: Each of the five parallelohedra admits a minimal (and weakly neighborly) model, moreover in three cases this is unique up to natural equivalence. Altogether there are 14 inequivalent minimal models.