  In this paper the totally asymmetric exclusion process (TASEP) with parallel
update on an open lattice of size $L$ is considered in the maximum-current
region. A formal expression for the generating function for the weight of
configurations with $N$ particles is given. Further an interpretation in terms
of $(u,l,d)$-colored weighted Motzkin paths is presented. Using previous
results (Woelki and Schreckenberg 2009 {\it J. Stat. Mech} P05014, Woelki 2010
{\it Cellular Automata}, pp 637-645) the generating function is compared with
the one for a possible 2nd-class particle dynamics for the parallel TASEP. It
is shown that both become physically equivalent in the thermodynamic limit.
