This paper examines a novel stars-parallel fiber laser synchronization local area network (LAN) and its hyperchaos synchronization systems. Its network structure was inducted, and its stars were composed of three-ring lasers of an erbium-doped fiber three-ring laser while its network nodes were composed of two types of nodes from two different sets of erbium-doped fiber dual-ring lasers; together, these nodes formed two groups in two links, each made up of two types. The paper illustrates our physical models of the network, including the stars, the nodes, and the parallel synchronization. The network synchronization structure and the laser synchronization cluster are presented. The system has the characteristics of star networks, parallel networks, multi-node networks. Our numerical results demonstrated that two erbium-doped fiber three-ring laser rings can synchronize with two different types of two sets of erbium-doped fiber dual-ring lasers in two links using the anti-phase-drive-response and delayed-time-feedback methods, respectively; this result included stars-parallel synchronizations in two links, the stars synchronizing with the nodes, parallel synchronization, hyperchaos synchronization, and existence of the laser synchronization cluster. Network functions indicated that hyperchaos in stars lead to certain behaviors in both the network and its nodes, and which will help in security and multi-channel secure communication. Our proposed fiber network has the potential to play a key reference role in the study of LAN, optical network, and secure communication.