A diagonal implicit scheme for solving quasilinear parabolic systems is presented. Every equation of the difference system can be independently solved on each time step and its solution absolutely converges to the accurate solution. Moreover, the algorithm is simple and is approable to MIMD or SIMD parallel computing with a higher degree of parallelism. The new scheme is constructed by adding an artificial viscosity for preserving the absolute convergence and a simple necessary condition about the coefficient of the artificial viscosity is derived.