Most recent schema matching systems combine multiple components, each of which employs a particular matching technique with several knobs. The multi- component nature has brought a tuning problem, that is to determine which components to execute and how to adjust the knobs (e.g., thresholds, weights, etc.) of these components for domain users. In this paper, we present an approach to automatically tune schema matching systems using genetic algorithms. We match a given schema S against generated matching scenarios, for which the ground truth matches are known, and find a configuration that effectively improves the performance of matching S against real schemas. To search the huge space of configuration candidates efficiently, we adopt genetic algorithms (GAs) during the tuning process. To promote the performance of our approach, we implement parallel genetic algorithms on graphic processing units (GPUs) based on NVIDIA's Compute Unified Device Architecture (CUDA). Experiments over four real-world domains with two main matching systems demonstrate that our approach provides more qualified matches over different domains.