Promoting green packaging production represents a crucial strategy for the packaging industry in its pursuit of sustainable development. This study constructs a three-party evolutionary game model involving suppliers, manufacturers, and brands to examine their strategic decision-making under various scenarios. Simulation and analysis yield three principal findings. First, the system initially begins at (0,0,0) and may transition to a manufacturer-dominated intermediate state—either (1,1,0) or (0,1,1)—before gradually stabilizing at the equilibrium point (1,1,1). Second, supply chain decision-making is influenced by both internal and external factors. Internal factors include penalty mechanisms, carbon trading allocation, and cooperative concessions, whereas external factors comprise consumer preferences for traceability and the environmental attributes of packaging. Specifically, suppliers are primarily driven by internal factors, manufacturers are predominantly influenced by external factors, and brands are impacted by a combination of both. Third, serving as the central node in the supply chain, manufacturers enable upstream and downstream integration through traceable production, refine cooperative concession mechanisms to enhance brand participation, and harness market signals to promote green transformation and co-production among suppliers. Therefore, the effective management of the green packaging supply chain necessitates the establishment and ongoing refinement of a tripartite active cooperation mechanism. Additionally, cultivating consumer preferences for traceability is essential for advancing the long-term sustainable development of the supply chain.