In order to address the power demand and voltage control challenges of a multi-energy network of electricity, heat, and gas coupled within an integrated energy system is an urban park, a distributed coordination methodology that incorporates voltage deviation control is proposed. First, operation models for local equipment and power flow coupling optimization in the electricity-gas-heat network are established. Then, a multi-objective day-ahead dispatch model and a local dispatch model for each agent are proposed, aiming to minimize both the overall operating cost and the mean voltage deviation. To achieve this, an asynchronous coordination approach based on preference prior expressions and the alternating direction method of multipliers (ADMM) is employed to enable distributed scheduling. The integrated energy system composed of a 14-node power grid, a 14-node heating network, and a 15-node gas network is taken as a simulation example, and the accuracy and practicability of the multi-objective programming method proposed are verified by comparing it with the existing multi-objective solutions and analyzing the Pareto front coordinates. Additionly, under the same calculation load distribution, the computational efficiency of the asynchronous coordination method is 16.6% higher than that of the synchronous method, which verifies the effectiveness of the proposed algorithm.