In this paper, we study a class of parallel machine scheduling problem with a single loading and multiple unloading servers. We show an application of this problem in a truck-scheduling context. We design a mixed-integer-programming formulation and use it to solve instances of various sizes with different characteristics. The obtained results reveal that a state-of-the-art commercial solver is only able to tackle instances of limited size. To solve larger instances, a general variable neighborhood search is proposed. Our comprehensive computational experiments demonstrate how the proposed metaheuristic can provide high-quality solutions in reasonable computing time (a few seconds). This type of performance is particularly relevant when considering operational problems that must be tackled on a daily basis. Finally, a sensitivity analysis is performed with respect to the number of trucks.