Why lattice constant (unit cell) is always a parallelepiped? Wikipedia defines lattice constant as physical dimension of unit cells in a crystal lattice. 

Unit cell is defined as: 


  The unit cell is defined as the smallest repeating unit having the
  full symmetry of the crystal structure. The geometry of the unit
  cell is defined as a parallelepiped.


Why that elementary pattern that repeats itself in crystals is always a parallelepiped? In real life such elementary repeating unit/part/block is very often not a parallelepiped at all!

Is this some abstraction and a "bounding parallelepiped" is meant - the one into which definitely fits whatever real elementary repeating pattern is?
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.