Community detection in networks has been studied extensively in the last decade. Many criteria, expressing the quality of the partitions obtained, as well as a few exact algorithms and a large number of heuristics have been proposed. The parsimony criterion consists in minimizing the number of edges added or removed from the given network in order to transform it into a set of disjoint cliques. Recently Zhang, Qiu and Zhang have proposed a weighted parsimony model in which a weight coefficient is introduced to balance the numbers of inserted and deleted edges. These authors propose rules to select a good value of the coefficient, use simulated annealing to find optimal or near-optimal solutions and solve a series of real and artificial instances. In the present paper, an algorithm is proposed for solving exactly the weighted parsimony problem for all values of the parameter. This algorithm is based on iteratively solving the problem for a set of given values of the parameter using a row generation algorithm. This procedure is combined with a search procedure to find all lowest breakpoints of the value curve (i.e., the weighted sum of inserted and deleted edges). Computational results on a series of artificial and real world networks from the literature are reported. It appears that several partitions for the same network may be informative and that the set of solutions usually contains at least one intuitively appealing partition.