Parallel join algorithms have received much attention in recent years due to the rapid development of massively parallel systems such as MapReduce and Spark. In the database theory community, most efforts have been focused on studying worst-case optimal algorithms. However, the worst-case optimality of these join algorithms relies on the hard instances having very large output sizes. In the case of a two-relation join, the hard instance is just a Cartesian product, with an output size that is quadratic in the input size. In practice, however, the output size is usually much smaller. One recent parallel join algorithm by Beame et al. has achieved output-optimality (i.e., its cost is optimal in terms of both the input size and the output size), but their algorithm only works for a 2-relation equi-join and has some imperfections. In this article, we first improve their algorithm to true optimality. Then we design output-optimal algorithms for a large class of similarity joins. Finally, we present a lower bound, which essentially eliminates the possibility of having output-optimal algorithms for any join on more than two relations.