Using numerical simulations, we study the asymmetric exclusion model with open boundaries, particlewise disorder and parallel dynamics. At each time step, particles are injected at the left boundary with probability , removed on the right with probability , and jump in the bulk with probability , where is a random rate associated with each injected particle . The parameter interpolates between fully parallel and random sequential dynamics. The phase diagram in the -plane displays high-density, low-density and maximum-current phases, with the first-order transition line between high- and low-density phases shifted away from the line . Within the low-density phase a platoon phase transition occurs, many features of which can be explained using exact results for asymmetric exclusion with particlewise disorder on the ring. In a certain region of parameter space the disorder induces a cusp in the current-density relation at maximum flow. Our simulations indicate that this does not affect the topology of the phase diagram, nor the familiar -decay of the density profile in the maximum-current phase.