The interdependency among the electricity, gas, heat, and cooling energy systems is ever-increasing. The flexible energy utilization patterns on the demand side and gas flow dynamics in the transmission system bring both opportunities and challenges to the reliable operation of integrated energy systems (IES). For example, if the electricity supply is interrupted, the gas system can ramp up the gas supply to the gas-fired units using linepacks. By this means, the reliability of the electricity system at this moment can be improved, while the gas system's capability of withstanding future risks may be undermined. Therefore, the operational reliability between different energy systems and time periods should be carefully balanced. This paper proposes an operational reliability evaluation framework for the IES considering flexibilities from both the demand side and transmission system. Firstly, the flexibilities of end-users and linepacks are explored based on the Energy Hub and gas flow dynamics models. Then, the reliability models of IES components are developed using the discretized-time Markov process to characterize the temporal state evolution in the operational horizon. A look-ahead contingency management scheme of the IES is then proposed to minimize the electricity and gas load curtailments. Taking account of all the possible system states, the operational reliabilities of the IES are evaluated using the time-sequential Monte Carlo simulation. Finally, the proposed method is validated by using the IEEE Reliability Test System and the practical Belgium gas transmission system.