Parallel evolutionary algorithms (PEAs) have been studied for reducing the execution time of evolutionary algorithms by utilizing parallel computing. An asynchronous PEA (APEA) is a scheme of PEAs that increases computational efficiency by generating a new solution immediately after a solution evaluation completes without the idling time of computing nodes. However, because APEA gives more search opportunities to solutions with shorter evaluation times, the evaluation time bias of solutions negatively affects the search performance. To overcome this drawback, this paper proposes a new parent selection method to reduce the effect of evaluation time bias in APEAs. The proposed method considers the search frequency of solutions and selects the parent solutions so that the search progress in the population is uniform regardless of the evaluation time bias. This paper conducts experiments on multi-objective optimization problems that simulate the evaluation time bias. The experiments use NSGA-III, a well-known multi-objective evolutionary algorithm, and compare the proposed method with the conventional synchronous/asynchronous parallelization. The experimental results reveal that the proposed method can reduce the effect of the evaluation time bias while reducing the computing time of the parallel NSGA-III.