In this note, we derive various convexity and monotonicity properties of performance measures in parallel GI/GI/1 queues with Bernoulli routing. We then use these results to establish that the equilikely assignment is optimal among all Bernoulli routing. The main tools are notions of stochastic convexity recently introduced by Shaked and Shanthikumar. The optimality results are couched in terms of the stochastic orderings st and icx.