In state-of-the-art autonomous vehicles, data from the vehicle's sensors is often processed using fast and expensive onboard hardware. Such an onboard processing scheme quickly drains the vehicle's battery and consumes computing resources. Recent research proposed to offload parts of processing tasks onto cloud. However, offloading tasks to the cloud is challenging because of the low latency needed for reliable and safe autonomous driving decisions. To address this issue, we propose an Age of Processing (AoP)-aware offloading mechanism for autonomous vehicles. First, we develop a collaboration space of edge clouds to process data closely as possible to the vehicles. Second, we reveal a new communication planning model that allows the vehicle to find suitable open radio units available in route to offload tasks to edge clouds and reduce variation in offloading delay. Third, we formulate an optimization problem that minimizes AoP, i.e., elapsed time from generating tasks and getting computation results. Our AoP-based approach allows a status update to be available to the vehicle after computation. To solve the formulated non-convex problem, we apply dual decomposition and design an AoP-aware algorithm to compute the solution in near real-time. The results demonstrate that our approach meets computation deadlines while minimizing AoP.