We address the numerical solution of Lyapunov, algebraic and differential Riccati equations, via the matrix sign function, on platforms equipped with general-purpose multi-core processors and, optionally, one or more graphics processing units (GPUs). In the paper, we review the solvers for these equations as well as the underlying methods, emphasizing their concurrency and scalability, and providing a few details on their parallel implementation. Our experimental results show that this class of hardware provides sufficient computational power to tackle large-scale problems, which only a few years ago would have required a cluster of computers.