The participation of third-party companies in the recycling of livestock and poultry breeding waste (LPBW) is an effective way to solve the pollution problems of small- and medium-scale livestock and poultry farming. This article constructs a dynamic differential game model for LPBW recycling based on the differential game, which includes small and medium-scale livestock and poultry farmers (SM-LPF), third-party companies, and local governments in the same analytical framework, and introduces corporate social responsibility into the model. Four waste recycling scenarios, namely the no cost-sharing contract scenario, the polluter-pays scenario, the third-party pays scenario, and the centralized decision-making scenario, are considered, and the four scenarios are solved, analyzed, and simulated. The results show that: (1) in the absence of government subsidies, both polluter-pays contracts and third-party pays contracts are effective in increasing system utility and achieving pareto improvements. (2) The polluter pays contract is established on the premise that the environmental liability coefficient of the SM-LPF needs to reach a certain threshold, and the third-party pays for the recycling contract is established on the condition that the spillover effect of cleaner production reaches a certain threshold. (3) Government subsidies can provide incentives and coordinate the interests of the waste recycling system, and the size of the optimal subsidy coefficients of the government for the SM-LPF and the third-party firms is related to the positions of the two parties in the structure of the game. This study provides an important theoretical basis and practical guidance for promoting the recycling of LPBW.