This paper discusses isomorphism between invertible torus parabolic maps and periodicity where periodic point sets in horocyclic case are dense in the torus topology, and some semi-rational cases possess periodic points of all periods. For the integral parabolic maps on the torus, the topological entropy is zero. Symbolic entropy and complexity of planar piecewise parabolic maps are investigated. Their difference under the plane topology and the torus topology respectively for the same torus parabolic maps is also discussed.