Membrane computing is an emergent research area studying the behavior of living cells to define bio-inspired computing devices, also called P systems. Such devices provide polynomial time solutions to NP-complete problems by trading time for space. The efficient simulation of P systems poses three major challenging issues: an intrinsic massive parallelism of P systems, an exponential computational workspace, and a non-intensive floating point nature. This paper analyzes the simulation of a family of recognizer P systems with active membranes that solves the satisfiability problem in linear time on three different architectures: a shared memory multiprocessor, a distributed memory system, and a manycore graphics processing unit (GPU). For an efficient handling of the exponential workspace created by the P systems computation, we enable different data policies on those architectures to increase memory bandwidth and exploit data locality through tiling. Parallelism inherent to the target P system is also managed on each architecture to demonstrate that GPUs offer a valid alternative for high-performance computing at a considerably lower cost. Our results lead to execution time improvements exceeding 310 and 78, respectively, for a much cheaper high-performance alternative.