This paper designs a distributed algorithm for solving sparse semidefinite programming (SDP) problems, based on the alternating direction method of multipliers (ADMM). It is known that exploiting the sparsity of a large-scale SDP problem leads to a decomposed formulation with a lower computational cost. The algorithm proposed in this work solves the decomposed formulation of the SDP problem using an ADMM scheme whose iterations consist of two subproblems. Both subproblems are highly parallelizable and enjoy closed-form solutions, which make the iterations computationally very cheap. The developed numerical algorithm is also applied to the SDP relaxation of the optimal power flow (OPF) problem, and tested on the IEEE benchmark systems.