  We consider the dynamics of a single shock in a partially asymmetric simple
exclusion process (PASEP) on a finite lattice with open boundaries in the
sublattice-parallel updating scheme. We then construct the steady state of the
system by considering a linear superposition of these shocks. It is shown that
this steady state can also be written in terms of a product of four non-
commuting matrices. One of the main results obtained here is that these
matrices have exactly the same generic structure as the matrices first
introduced in Jafarpour and Masharian (2009 Phys. Rev. E 79 051124) indicating
that the steady state of a one-dimensional driven-diffusive system can be
written as a linear superposition of product shock measures. It is now easy to
explain the two-dimensional matrix representation of the PASEP with parallel
dynamics introduced in Essler and Rittenberg (1996 J. Phys. A: Math. Gen. 29
3375) and Honecker and Peschel (1997 J. Stat. Phys. 88 319).
