The integration of parallel computing techniques into metaheuristics has traditionally represented a promising approach to tackle computationally demanding optimization problems. In the last years, metaheuristics have evolved by including more complex search mechanisms whose parallelization often leads to performance issues under classic parallel schemes. This work investigates the asynchronous non-generational parallelization model, which is aimed at dealing with performance pitfalls by allowing worker threads to behave as asynchronous independent agents. We incorporate asynchronous principles into a recently proposed metaheuristic for multiobjective optimization, the Indicator-Based Multiobjective Bat Algorithm, and apply the resulting approach to solve a real-world problem in the bioinformatics domain: the reconstruction of evolutionary histories. Experiments on multicore multiprocessor systems comprising up to 64 cores reveal the suitability of the model to address the main challenges of the metaheuristic design under study, outperforming other implementations and methods in terms of parallel performance while also achieving significant solution quality.