A regular octagon will not tessellate - the 'spaces' left over are squares.
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If the octagon is a regular octagon (all eight sides and all eight angles are equal), then it is a regular polygon. If it is not a regular octagon (i.e. if it is an irregular octagon) then is it an irregular polygon.
"All of the polyiamonds of order eight or less, with the exception of one of the heptiamonds will tessellate the plane. The exception is the V-shaped heptiamond."Source: http://www.mathpuzzle.com/Tessel.htm
Parallelograms have only four sides. Polygons with eight sides are octagons.
Whether it is a regular or irregular octagon does not matter. An octagon is, by definition, a 2-dimensional figure with eight sides, and therefore with eight angles. A regular octagon will have eight equal angles and an irregular octagon will have eight angles some of which are not equal to the others.