Our study contributes to the debate on the evolution of cooperation in the single-shot Prisoner's Dilemma (PD) played on networks. We construct a model in which individuals are connected with positive and negative ties. Some agents play sign-dependent strategies that use the sign of the relation as a shorthand for determining appropriate action toward the opponent. In the context of our model in which network topology, agent strategic types and relational signs coevolve, the presence of sign-dependent strategies catalyzes the evolution of cooperation. We highlight how the success of cooperation depends on a crucial aspect of implementation: whether we apply parallel or sequential strategy update. Parallel updating, with averaging of payoffs across interactions in the social neighborhood, supports cooperation in a much wider set of parameter values than sequential updating. Our results cast doubts about the realism and generalizability of models that claim to explain the evolution of cooperation but implicitly assume parallel updating.