In this work, we propose APM-MOEA, a parallel model to efficiently solve the Multi-Objective Quadratic Assignment problem (MQAP) using an evolutionary algorithm. It is based on an island model with objective space division. Its main features are a global view from an organizer to achieve better distribution of solutions, an asynchronous communication scheme to reduce parallel overhead, control islands to improve diversity and a local search procedure to improve the quality of solutions. Extensive experiments have been conducted using the GISMOO algorithm to compare APM-MOEA with a recent specialized parallel algorithm and two state-of-the-art island-based models in the resolution of the MQAP. Results show that according to four multi-objectives metrics, APM-MOEA outperforms all implementations in terms of convergence or diversity.