Why lattice constant (unit cell) is always a parallelepiped?
A unit cell is always a parallelepiped because this is the definition of a unit cell. In case when you don't have a repeating unit of this kind (which might indeed be the case in real life), you don't have a crystal, also by definition. You might be dealing with a quasicrystal or something.

3D periodicity is what defines a crystal. If you have it, this means you have three non-coplanar periodic vectors, and these inevitably define a parallelepiped.