For robotics, multi-agent cooperative motion planning (MACMP) is an important but complex problem since it contains highly coupled dynamics and collision-free constraints. In this letter, a general parallel solution framework is proposed based on alternating direction method of multipliers (ADMM), which decouples the complicated coupling constraints and decomposes original problem into independent sub-problems for each agent, thereby making the iterative computation of ADMM and the solving of sub-problems parallel. The convergence of ADMM is analyzed, and the simulation results verify that proposed framework can better balance solution optimality and computational efficiency, compared with interior point method (IPM) and progressively constrained dynamic optimization algorithm (PCDO).