The utilization of battery energy storage systems (BESS) in vehicle-to-grid (V2G) and plug-in hybrid electric vehicles (PHEVs) benefits the realization of net-zero in the energy-transportation nexus. Since BESS represents a substantial part of vehicle total costs, the mitigation of battery degradation should be factored into energy management strategies. This paper proposes a two-stage BESS aging quantification and health-aware energy management method for reducing vehicle battery aging costs. In the first stage, a battery aging state calibration model is established by analyzing the impact of cycles with various Crates and depth of discharges based on a semi-empirical method. The model is further linearized by learning the mapping relationship between aging features and battery life loss with a linear-in-the-parameter supervised learning method. In the second stage, with the linear battery life loss quantification model, a neural hybrid optimization-based energy management method is developed for mitigating vehicle BESS aging. The battery aging cost function is formulated as a linear combination of system states, which simplifies model solving and reduces computation cost. The case studies in an aggregated EVs peak-shaving scenario and a PHEV with an engine-battery hybrid powertrain demonstrate the effectiveness of the developed method in reducing battery aging costs and improving vehicle total economy. This work provides a practical solution to hedge vehicle battery degradation costs and will further promote decarbonization in the energy-transportation nexus.