  Using the Bethe ansatz we obtain the determinant expression for the time
dependent transition probabilities in the totally asymmetric exclusion process
with parallel update on a ring. Developing a method of summation over the roots
of Bethe equations based on the multidimensional analogue of the Cauchy residue
theorem, we construct the resolution of the identity operator, which allows us
to calculate the matrix elements of the evolution operator and its powers.
Representation of results in the form of an infinite series elucidates
connection to other results obtained for the ring geometry. As a byproduct we
also obtain the generating function of the joint probability distribution of
particle configurations and the total distance traveled by the particles.
