The efficient utilization of integrated demand response (IDR) is substantially impeded by the centralized management structure, where the integrated energy operator (IEO) exclusively acquires IDR resources from integrated load aggregators (ILAs). However, as a unique commodity, the demand response transactions among ILAs suffer from insufficient attention. In this paradigm, the potential of IDR to enhance system flexibility is restricted, resulting in supply-demand imbalance and reduced economic efficiency across the entire system. Therefore, it is imperative to utilize IDR resources in decentralized manner and facilitate the synergy between IEO and ILAs. To this end, a cooperative demand response management (DRM) framework for integrated energy system (IES) is proposed. Firstly, the decentralized DR market is built by integrating peer-to-peer (P2P) transactions with demand response exchange (DRX) mechanism, in which Nash bargaining theory and alternating direction method of multipliers (ADMM) algorithm are applied for optimal interest allocation and privacy preservation, respectively. Secondly, the price elasticity matrix of gas-electricity, which depicts cross-price elasticity of gas demand with respect to electricity in retail market and vice versa, is introduced to furtherly exploit the flexibility of demand side resource. Thirdly, the cooperative DRM framework is established, where the P2P market synergizes with the traditional retail market to foster the efficient utilization of IDR resources. Case study demonstrate that the proposed DRM framework effectively harnesses the potential of IDR, and facilitates a win-win situation between the ILAs and IEO.