This paper aims to develop a fast, parallelizable algorithm for an arbitrary decomposable semidefinite program (SDP). To formulate a decomposable SDP, we consider a multi-agent canonical form represented by a graph, where each agent (node) is in charge of computing its corresponding positive semidefinite matrix subject to local equality and inequality constraints as well as overlapping (consistency) constraints with regards to the agent's neighbors. Based on the alternating direction method of multipliers, we design a numerical algorithm, which has a guaranteed convergence under very mild assumptions. Each iteration of this algorithm has a simple closed-form solution, consisting of matrix multiplications and eigenvalue decompositions performed by individual agents as well as information exchanges between neighboring agents. The cheap iterations of the proposed algorithm enable solving real-world large-scale conic optimization problems.