We study the maximum set coverage problem in the massively parallel model. In this setting, m sets that are subsets of a universe of n elements are distributed among m machines. In each round, these machines can communicate with each other, subject to the memory constraint that no machine may use more than \(\tilde{O} \left( n \right) \) memory. The objective is to find the k sets whose coverage is maximized. We consider the regime where \(k = \Omega (m)\) (i.e., \(k = m/100\) ), \(m = O(n)\) , and each machine has \(\tilde{O} \left( n \right) \) memory \(^1\) . Maximum coverage is a special case of the submodular maximization problem subject to a cardinality constraint. This problem can be approximated to within a \(1-1/e\) factor using the greedy algorithm, but this approach is not directly applicable to parallel and distributed models. When \(k = \Omega (m)\) , to obtain a \(1-1/e-\epsilon \) approximation, previous work either requires \(\tilde{O} \left( mn \right) \) memory per machine which is not interesting compared to the trivial algorithm that sends the entire input to a single machine, or requires \(2^{O(1/\epsilon )} n\) memory per machine which is prohibitively expensive even for a moderately small value \(\epsilon \) . Our result is a randomized \((1-1/e-\epsilon )\) -approximation algorithm that uses \(\begin{aligned} O(1/\epsilon ^3 \cdot \log m \cdot (\log (1/\epsilon ) + \log m)) \end{aligned}\) rounds. Our algorithm involves solving a slightly transformed linear program of the maximum coverage problem using the multiplicative weights update method, classic techniques in parallel computing such as parallel prefix, and various combinatorial arguments.