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Numerical schemes for solving the Vlasov-Maxwell system equations are studied. For studies of cross-scale coupling between fluid-scale dynamics and particle-scale kinetics in collisionless plasma, both highly scalable kinetic code and huge supercomputer are essential. In the present study, a new parallel Vlasov-Maxwell solver is developed by adopting a stable but less dissipative scheme for time integration of conservation laws. The new code has successfully achieved a high scalability on massively parallel supercomputers with multicore scalar processors. The new code has been applied to 2P3V (two dimensions for position and three dimensions for velocity) problems of cross-scale plasma processes such as magnetic reconnection, Kelvin-Helmholtz instability, and interaction between the solar wind and an asteroid.