-We have developed a discretization method of an arbitrarily given order of accuracy with respect to the time discretization parameter for the first-order evolution differential equation in Banach space. The method includes two levels of parallelism: the operator exponential needed for the calculation of the evolution operator can be computed in parallel analogously to [6] (inner parallelism) and then we can compute in parallel the evolution operator at various points of the time mesh (outer parallelism).2000 Mathematics Subject Classification: 65F50; 65F30; 15A09; 15A24; 15A99.