Description Logic (DL) ontologies underpin many Semantic Web applications. Consequence-based reasoning, which integrates techniques from hypertableau and resolution, has proved effective for tasks such as consistency checking and classification in both lightweight and expressive DLs. However, existing reasoners often fall short when applied to large, complex ontologies commonly found in domains such as healthcare and industry. In this paper, we extend the state-of-the-art consequence-based reasoner Sequoia [13] to support parallel reasoning, improving its scalability by leveraging system architectures with multiple cores. We explore and evaluate two parallelisation strategies for consequence-based reasoners: message passing and thread pools, and demonstrate their application within the Sequoia reasoner. Our extensive empirical evaluation shows that thread pool-based implementations achieve superior performance and resource efficiency, offering up to 2.62x speedup over the baseline on hard ontologies. We also explore the effect of increasing the number of available cores or restricting the expressivity of the ontology in the performance of our implementations.