Parity games are two-player zero-sum games of infinite duration played on finite graphs for which no solution in polynomial time is still known. Solving a parity game is an $$\text{ NP }\cap \text{ co-NP }$$ NP ∩ co-NP problem, with the best worst-case complexity algorithms available in the literature running in quasi-polynomial time. Given the importance of parity games within automated formal verification, several practical solutions have been explored showing that considerably large parity games can be solved somewhat efficiently. Here, we propose a new approach to solving parity games guided by the efficient manipulation of a suitable matrix-based representation of the games. Our results show that a sequential implementation of our approach offers very competitive performance, while a parallel implementation using GPUs outperforms the current state-of-the-art techniques. Our study considers both real-world benchmarks of structured games as well as parity games randomly generated. We also show that our matrix-based approach retains the optimal complexity bounds of the best recursive algorithm to solve large parity games in practice.