The paper presents parallel algorithms for calculating the exact value of all-terminal reliability of a network with unreliable edges and absolutely reliable nodes. A random graph is used as a model of such network. The algorithms are based on the factorization procedure which is a well-known sequential method of a reliability calculation. Parallelization of the algorithms consists in sending subgraphs, arising during the factorization of a network on the master process, to the rest of the processes that perform sequential calculation by the factorization. The basic idea of the parallel algorithms proposed is to distinguish those subgraphs, arising during the factorization in the work processes, that are relatively hard for reliability calculation. These subgraphs are sent back to the master process, which runs the computation of their reliability in a recursive way, i.e. according to the same scheme as with an initial graph. The results of the numerical experiments are given.