With the goal of carbon emission peak and carbon neutrality, it is of great significance for energy conservation and emission reduction to design a reasonable carbon trading mechanism so that the source, grid and load can participate in the carbon market together. In this context, a distributed collaborative low-carbon economic dispatching strategy of source, grid and load considering the dual-layer carbon trading mechanism with reward and punishment is proposed. First, a collaborative low-carbon dispatching model of integrated energy suppliers and park under the dual-layer carbon trading mechanism with reward and punishment is established. The integrated energy suppliers directly participate in the external ladder-type carbon trading market with reward and punishment, and the parks indirectly participate in the carbon market by paying carbon fees to integrated energy suppliers or obtaining carbon benefit, thus stimulating all entities to actively participate in energy conservation and emission reduction. Secondly, a ladder-type carbon price model with reward and punishment is established, and a carbon cost/benefit allocation method of integrated energy suppliers and multi-parks under the ladder-type carbon price mechanism with reward and punishment is proposed to ensure the effectiveness and rationality of carbon cost/benefit allocation. Thirdly, the low-carbon dispatching models for the integrated energy suppliers and multi-parks are established, respectively, and the cooperation game between parks is described based on Nash bargaining, so as to reduce carbon emissions and improve social benefits through mutual power exchange among parks. Then, a dual-layer distributed solution for nested alternating direction method of multipliers based on adaptive adjustment mechanism is proposed. Finally, the source-grid-load integrated energy system composed of IEEE 14-bus distribution network and 12-line thermal network is taken as a case to verify the effectiveness of the proposed model and method. This work is supported by National Key R&D Program of China (No. 2022YFB3304502). © 2024 Automation of Electric Power Systems Press. All rights reserved.