Recent years have witnessed numerous technical breakthroughs in connected and autonomous vehicles (CAVs). On the one hand, these breakthroughs have significantly advanced the development of intelligent transportation systems (ITSs); on the other hand, these new traffic participants introduce more complex and uncertain elements to ITSs from the social space. Digital twins (DTs) provide real-time, data-driven, precise modeling for constructing the digital mapping of physical-world ITSs. Meanwhile, the metaverse integrates emerging technologies such as virtual reality/mixed reality, artificial intelligence, and DTs to model and explore how to realize improved sustainability, increased efficiency, and enhanced safety. More recently, as a leading effort toward general artificial intelligence, the concept of foundation model was proposed and has achieved significant success, showing great potential to lay the cornerstone for diverse artificial intelligence applications across different domains. In this article, we explore the big models embodied foundation intelligence for parallel driving in cyber-physical-social spaces, which integrate metaverse and DTs to construct a parallel training space for CAVs, and present a comprehensive elucidation of the crucial characteristics and operational mechanisms. Beyond providing the infrastructure and foundation intelligence of big models for parallel driving, this article also discusses future trends and potential research directions, and the “6S” goals of parallel driving.