In face of the increasing penetration of renewable energy, compressed air energy storage (CAES) is promising in improving the flexibility of the conventional coal-fired combined heating and power plant (CHPP). In order to further broaden the operational range and improve the overall efficiency, this paper introduces the molten salt heat storage (MSHS) into the CHPP-CAES system. Based on the thermodynamic model, the effect of the MSHS is discussed in terms of the energy efficiency, exergy efficiency, and the power regulation based on the topology with and without MSHS. The inclusion of the MSHS can improve the exergy efficiency and reduce the average coal consumption for power supply, while also effectively widening the power regulation range. Parameter analysis is carried out in terms of the compression and expansion stages of CAES, the steam extraction position of MSHS, throttle valve pressure drop, and the capacities of CAES and MSHS. It is revealed that, extracting steam from the boiler reheater and reducing the CAES capacity can lead to better efficiency but smaller operational range. Furthermore, reducing the throttle valve pressure drop and improving the MSHS capacity are simultaneously favorable for the overall performance and operating range. These conclusions can provide theoretical guidance for the structure and parameter design of the CHPP in hybrid with CAES and MSHS.