New mission concepts are increasingly considering the use of ion propulsion for fuel-efficient navigation in deep space. The development of new low-thrust mission concepts requires efficient methods to rapidly determine feasibility and thoroughly explore trade spaces. This paper presents parallel, evolutionary computing methods to assess a 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 approximate optimal solutions. The methods are coupled with the Primer Vector theory, where a 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 mission scenarios: i) an orbit transfer around Earth and ii) a transfer between two distant retrograde orbits around Europa. The solutions found with the present methods are comparable to those obtained by other state-of-the-art trajectory optimizers. The required computational time can be up to several orders of magnitude shorter than that of other optimizers thanks to the utilization of the distributed computing environment, the significant reduction of the search space dimension with the Primer Vector theory, and the efficient and synergistic exploration of the remaining search space with evolutionary computing.