Despite technological advances, vehicles are still unable to meet the demands of some applications for massive computational resources in a feasible time. One way to deal with this situation is to integrate the computation offloading technique into a vehicular edge computing system. This integration allows application tasks to be executed on neighboring vehicles or edge servers coupled to base stations. However, the dynamic nature of vehicular networks, allied to over- loaded servers, can lead to failures and reduce the effectiveness of the offloading technique. Therefore, we propose a context-oriented framework for computation offloading to reduce the application execution time and maintain high reliability in vehicular edge computing. The framework modules perform computational resources discovery, contextual data gathering, computation tasks distribution, and failure recovery. Its main part is a task assignment algorithm that seeks the best possible server to execute each application task, using contextual information and WAVE and 5G networks. The results of extensive experiments in different vehicular environments show that our framework reduces up to 70.3% of total execution time compared to totally local execution and up to 42.9% compared to other literature approaches. Concerning reliability, our framework achieves to offload up to 89.4% of all tasks and needs to recover only 0.8% of them. Thus, our solution outperforms the totally local execution of the application and other existing computation offloading solutions.