A new approach to polymer simulation well suited to massively parallel architectures is presented. The approach is based on a novel two-space algorithm that enables 50% of the monomers to be updated in parallel. The simplicity of this algorithm enables implementation and comparison of different platforms. Such comparisons are relevant to a wide variety of scientific applications. We tested this algorithm on three commercially available machines, the MP-1, KSR1, and CM-2; and on a prototype of the CAM-8 architecture. Among the commercial machines we found the MP-1 provided the best performance for highly-parallel fine-grained simulations. Effective utilization of the KSR1 was achieved with attention to synchronization requirements. The small (8 node) CAM-8 prototype, with a kind and cost of hardware comparable to an engineering workstation, achieved a performance within a factor of two of the MP-1 for our application.