The development of new mission concepts requires efficient methodologies to analyze, design, and simulate the concepts before implementation. New mission concepts are increasingly considering the use of ion thrusters for fuel-efficient navigation in deep space. This paper presents parallel, evolutionary computing methods to design trajectories of spacecraft propelled by ion thrusters and assesses the trade-off between delivered payload mass and required flight time. The developed methods utilize a distributed computing environment in order to speed up computation, and use evolutionary algorithms to find globally Paretooptimal solutions. The methods are coupled with two main traditional trajectory design approaches, which are called direct and indirect. In the direct approach, thrust control is discretized in either arc time or arc length, and the resulting discrete thrust vectors are optimized. In the indirect approach, the thrust control problem is transformed into a co-state control problem and the initial values of the co-state vector are optimized. The developed methods are applied to two problems: 1) an orbit transfer around the Earth and 2) a transfer between two distance retrograde orbits around Europa, the icy Galilean moon closest to Jupiter. The optimal solutions found with the present methods are comparable to other state-of-the-art trajectory optimizers, while the required computation time is often several orders of magnitude less thanks to an intelligent design of control vector discretization, advanced algorithmic parameterization, and parallel computing.