Shared bicycles parking is one of the most important problems in urban transportation management. On the surface, it is caused by the irregular parking behavior of shared bicycles users and the untimely management and dispatching of operating enterprises. In essence, it is the result of the lack of effective collaborative governance mechanism, the unclear division of parking rights and responsibilities, the imbalance of cost sharing among the stakeholders of shared bicycles. Considering the bounded rationality of the government and bike-sharing enterprises, this study establishes an quantal response equilibrium (QRE) model for extensive form games to analyze the parking management game mechanism, and uses Gambit to simulate and analyze a numerical example. The research indicates that game results of two players gradually tend to the subgame-perfect equilibrium based on complete rationality with the increase of the degree of rationality. The government's grasp of more information about the operation of enterprises will help to achieve the game result of maximizing social utility. The government can reduce its own financial expenditure and total social cost by introducing effective punishment measures or taking charge of parking management. Quantifying the punishment standard can promote bike-sharing enterprises to take responsibility for parking management. In addition, the factors including the degree of bounded rationality, the number of illegal parking, operation efficiency of incentive mechanism and technological methods will affect the behaviour strategy of parking management.