  We study the one-dimensional discrete time totally asymmetric simple
exclusion process with parallel update rules on a spatially periodic domain. A
multi-point space-time joint distribution formula is obtained for general
initial conditions. The formula involves contour integrals of Fredholm
determinants with kernels acting on certain discrete spaces. For a class of
initial conditions satisfying certain technical assumptions, we are able to
derive large-time, large-period limit of the joint distribution, under the
relaxation time scale $t=O(L^{3/2})$ when the height fluctuations are
critically affected by the finite geometry. The assumptions are verified for
the step and flat initial conditions. As a corollary we obtain the multi-point
distribution of discrete time TASEP on the whole integer lattice $\mathbb{Z}$
by taking the period $L$ large enough so that the finite-time distribution is
not affected by the boundary. The large time limit for multi-time distribution
for discrete time TASEP on $\mathbb{Z}$ is then obtained for the step initial
condition.
