Mobile-edge computing (MEC) has been recognized as a promising solution to provide efficient communication and computation capabilities for mobile users (MUs). However, the problem relating to parallel offloading and load balancing with multiple cooperative MEC servers and massive delay-sensitive execution workloads remains to be investigated. In this paper, we study a joint parallel offloading and load balancing policy for such an MEC system. We formulate a long-term system average-cost (i.e., weighted sum of energy consumption and execution delays) stochastic programming problem under MU battery level stability and delay constraints. Our aims include optimizing the data sizes for uplink offloading transmissions for optimized communication and computation resource allocation among multi-users, and maximizing the computation capability utilization of cooperative MEC servers by load balancing. To solve this problem, we first design a Lyapunov-based centralized cost management algorithm (LYP-CCMA) to obtain the optimal system average-cost under the battery level stability constraints. Further, we propose two algorithms based on alternating direction method of multipliers (ADMM) to implement distributed resources allocation. Simulation results verify our analysis and demonstrate the superior performance of our proposed schemes over several baseline schemes.