We propose a distributed routing system for multi-robot cooperative transportation using parallel multiple processors with asynchronous data exchange. In the proposed method, each robot generates a routing to minimize each objective function comprising of traveling time and the penalty for violating constraints of cooperation with other robots. A near optimal routing is generated by repeating the generation of each routing and data exchange among the robots. Asynchronous data exchange is adopted to reduce the computation time. In order for the solution not to be trapped in a local optimum, the weighting factors violating the constraints are updated with a time-dependent method during parallel computation. The proposed method is applied to motion planning for cooperative transportation of an experimental 4 real robots system. The effectiveness of the proposed method is investigated on various types of experimental conditions.