Computer simulations of experimentally comparable system sizes in soft matter often require considerable elapsed times. The use of many cores can reduce the needed time, ideally proportionally to the number of processors. In this paper a parallel computational method using coarray Fortran is implemented and tested for large systems of purely block copolymer melts, as well as block copolymer nanocomposites. A satisfactory strong scaling is shown up to 512 cores while a weak scaling with a drop in performance is achieved up to 4096 cores. The scaling of the parallel cell dynamic simulations scheme displays no drawbacks over MPI and provides an example of the simplicity of the coarray approach. The code has been tested on several architectures and compilers. The hybrid block copolymer/nanoparticle algorithm can achieve previously unavailable system sizes.