It is difficult to fully utilize the parallelism of large-scale computing environments in alpha-beta search. The naive parallel execution of subtrees would result in much less task pruning than may have been possible in sequential execution. This may even degrade total performance. To overcome this difficulty, we propose a two-level task scheduling policy in which all tasks are classified into two priority levels based on the necessity for their results. Low priority level tasks are only executed after all high priority level tasks currently executable have started. When new high priority level tasks are generated, the execution of low priority level tasks is suspended so that high level tasks can be executed. We suggest tasks be classified into the two levels based on the Young Brothers Wait Concept, which is widely used in parallel alpha-beta search. The experimental results revealed that the scheduling policy suppresses the degradation in performance caused by executing tasks whose results are eventually found to be unnecessary. We found the new policy improved performance when task granularity was sufficiently large.