In this paper, we describe the orbital structure of homogeneous fuzzy parallel dynamical systems on Zadeh operators. In particular, we show that, when the (global) evolution operator is either the Zadeh conjunction operator AND or the Zadeh disjunction operator OR, all the points of the system are fixed points or eventually fixed points. Specifically, such systems present infinitely many fixed points. On the other hand, when the (global) evolution operator is either the Zadeh operator NOR or the Zadeh operator NAND, the orbital structure consists of an only fixed point and infinitely many 2-periodic points.