Flexible Job Shop Scheduling is one of the most difficult optimization problems known. In addition, modern production planning and control strategies require continuous and process-parallel optimization of machine allocation and processing sequences. Therefore, this paper presents a new method for process parallel Flexible Job Shop Scheduling using the concept of quantum computing based optimization. A scientific benchmark and the application to a realistic use-case demonstrates the good performance and practicability of this new approach. A managerial insight shows how the approach for process parallel flexible job shop scheduling can be integrated in existing production planning and control IT-infrastructure.