I tried using a scheduler. Problem is I could not find a simple blocking scheduler that just executes all tasks given to it and return afterwards. Using for example a threadpertask scheduler, I have to wait for the final task to be executed in the test thread. Using a scheduler in this test introduces a lot of overhead and problems. I'd much rather leave it this way and not have to deal with the complexity of scheduling. Also if the scheduling itself breaks, this test case will break. This is bad practice for unit testing.
Unless there is a scheduler that does exactly what my code does above, I prefer leaving my code the way it is. 
In order to solve this kerfuffle, we could talk about introducing a helper method for executing a bunch of tasks without a scheduler. That method could also be used in RequestParseTask for high priority task execution.