This work presents a multi-objective approach to solve the Container Loading Problem. The single-objective formulation of the problem has been widely studied in the related literature, trying to optimise the total volume of the packed pieces into the container. However, a rather common aspect in the scope of this problem is the weight limit of the containers, since they normally cannot exceed a certain weight for their transportation, and they should make the most without exceeding that limit. For this reason, we have focused on a multi-objective formulation of the problem which seeks to maximize the volume at the same time that the weight, without exceeding the weight limit. To solve this multi-objective problem we have applied multi-objective optimisation evolutionary algorithms given their great effectiveness with other types of real-world multi-objective problems. One of the goals of this work is to improve the results of the only known work in the literature that addresses the same problem with multiple objectives. Once we have achieved it, we have parallelized the problem applying different island-based models to enhance the effectiveness and efficiency of our approach.