Massively parallel join algorithms have received much attention in recent years, while most prior work has focused on worst-optimal algorithms. However, the worst-case optimality of these join algorithms relies on hard instances having very large output sizes, which rarely appear in practice. A stronger notion of optimality is output-optimal, which requires an algorithm to be optimal within the class of all instances sharing the same input and output size. An even stronger optimality is instance-optimal, i.e., the algorithm is optimal on every single instance, but this may not always be achievable. In the traditional RAM model of computation, the classical Yannakakis algorithm is instance-optimal on any acyclic join. But in the massively parallel computation (MPC) model, the situation becomes much more complicated. We first show that for the class of r-hierarchical joins, instance-optimality can still be achieved in the MPC model. Then, we give a new MPC algorithm for an arbitrary acyclic join with load O (IN over p + sqrtIN cdot OUT over p), where IN,OUT are the input and output sizes of the join, and p is the number of servers in the MPC model. This improves the MPC version of the Yannakakis algorithm by an O (sqrtOUT over IN ) factor. Furthermore, we show that this is output-optimal when OUT = O(p cdot IN), for every acyclic but non-r-hierarchical join. Finally, we give the first output-sensitive lower bound for the triangle join in the MPC model, showing that it is inherently more difficult than acyclic joins.