This work studies the crossroad traffic model with low velocity vehicles,wherein the model consists of two one-lane roads. The parallel updating rule without traffic lights is obeyed by all the cars in our model. Based on the principle for constructing phase diagrams and the local occupation probability method,the phase diagrams are constructed,and in every phase diagram,the formulas of current in each region can be obtained. The deterministic FI model with the periodic boundary is applied in our numerical simulations. It turns out that the numerical results are in exact accordance with the theoretical analysis. Besides,the parallel updating rule is more common in the real crossroad traffic and thus our work may be helpful for the study of traffic management.