The main goal of distribution network (DN) expansion planning is essentially to achieve minimal investment constrained by specified reliability requirements. The reliability-constrained distribution network planning (RcDNP) problem can be cast as an instance of mixed-integer linear programming (MILP) which involves ultra-heavy computation burden especially for large-scale DNs. In this paper, we propose a parallel computing based solution method for the RcDNP problem. The RcDNP is decomposed into a backbone grid and several lateral grid problems with coordination. Then, a parallelizable augmented Lagrangian algorithm with acceleration method is developed to solve the coordination planning problems. The lateral grid problems are solved in parallel through coordinating with the backbone grid planning problem. Gauss-Seidel iteration is adopted on the subset of the convex hull of the feasible region constructed by decomposition. Under mild conditions, the optimality and convergence of the proposed method are proven. Numerical tests show that the proposed method can significantly reduce the solution time and make the RcDNP applicable for real-world problems.