In order to study the optimal clearing problem of consumers in a fully decentralized peer-to-peer (P2P) energy market, this paper focused on solving the challenges of cooperation among consumers and maximizing social welfare in P2P markets. A new parallel and distributed alternating direction method of multipliers (ADMM) is adopted to derive the trading mechanism of P2P market. This method considers the utility function of each prosumer, and introduces a distributed generator (DG) and a battery energy storage system (BESS). In the algorithm, each prosumer synchronously exchanges a small amount of information with its neighboring prosumers through iteration and optimizes to meet different requirements. The effectiveness of the proposed method is demonstrated by numerical verification of the 6-peers system. Compared with the pool-based trading mechanism, the fully decentralized P2P problem increases the transaction power by 160% per unit time, and the social welfare increases from -9.47 yuan to 32.43 yuan.