This paper presents a randomized parallel algorithm for the Maximal Independent Set problem. Our algorithm uses a BSP-like computer with p processors and requires (n+m)/p larger or equal to p for a graph with n vertices and m edges. Under this scalability assumption, and after a preprocessing phase, it computes a maximal independent set after O(log p) communication rounds, with high probability, each round requiring linear computation time O((n+m)/p). The preprocessing phase is deterministic and important in order to ensure that degree computations can be implemented efficiently. For this, we give an optimal parallel BSP/CGM algorithm to the p-quantiles search problem, which runs in O(m log p/p) time and a constant number of communication rounds, and could be of interest in its own right, as shown in the full text.