As a typical aggregation platform, the regional integrated energy system (RIES) is required to pre-submit its flexibility for each time period when participating in demand response. However, existing studies fail to quantify the flexibility parameters necessary for RIES pre-reporting. Therefore, this paper introduces a flexibility quantification framework tailored for RIES engagement in demand response. The core principle of the framework is grounded in the RIES day-ahead dispatch model considering virtual energy storage, which transforms the flexibility quantification problem into dispatch optimization problems under baseline scenarios and flexibility invocation scenarios. In the flexibility invocation scenario, a control term is incorporated into the objective function to steer the system towards fulfilling flexibility requirements in the most economical strategy. The deviation of power and operating costs in both scenarios defines the system's flexibility and flexibility invocation cost, respectively. This framework holds broad applicability across various aggregation platforms and can measure flexibility across different time scales by adjusting the control interval of the day-ahead dispatch optimization model. Finally, the framework is validated through case studies, revealing the impact characteristics of system type and capacity on flexibility, and analyzing the sensitivity of flexibility to various uncertainty parameters.