Not every simple polygon can be tiled with parallelograms.

> Kannan, Sampath, and Danny Soroker. "Tiling polygons with parallelograms." *Discrete & Computational Geometry* 7, no. 2 (1992): 175-188. 
[DOI](https://doi.org/10.1007/BF02187834).

> Kenyon, Richard. "Tiling a polygon with parallelograms." *Algorithmica* 9, no. 4 (1993): 382-397.
[DOI](https://doi.org/10.1007/BF02187834).

Here are two untilable examples from the Kannan et al. paper:

     [![UnTilable][1]][1]


  [1]: https://i.sstatic.net/BVcKk.jpg