This paper proposes a multi-objective optimal energy management framework for the integrated electrical and natural gas network (IEGN) with combined cooling, heat, and power (CCHP) plants. Various energy conversion devices that are installed in the CCHP plant provide redundant generation options and energy pathways, which could be optimally chosen and shifted with given objectives, while meeting the multi-energy (ME) demands. However, this flexible energy dispatch manners may frequently change the energy distribution in the IEGN and challenge their mutual accommodation. In particular, the linepack reserve in the natural gas network, which supports the ramping capabilities of both the gas turbines and the flexible energy dispatch of the gas-dependent ME devices, is highly influenced. Without enough linepack reserve, not only will the flexible operation of the CCHP plants be hindered, but also the gas turbines will be prevented from balancing the supply and the demand in the electrical network, thus threatens the safety of the IEGN. Owing to this, the linepack reserve is modelled and jointly considered in the proposed energy management framework. The multi-objective optimization model that is proposed in this paper could simultaneously promote the economic benefits, safety, and efficiency of the IEGN, and Elitist Non-dominated Sorting Genetic algorithm II is used to solve it. At last, case studies demonstrate the effectiveness of the proposed method.