This paper presents a two-sided economic search model in which agents are searching for beneficial pairwise partnerships. In each search stage, each of the agents is randomly matched with several other agents in parallel, and makes a decision whether to accept a potential partnership with one of them. The distinguishing feature of the proposed model is that the agents are not restricted to maintaining a synchronized (instantaneous) decision protocol and can sequentially accept and reject partnerships within the same search stage. We analyze the dynamics which drive the agents' strategies towards a stable equilibrium in the new model and show that the proposed search strategy weakly dominates the one currently in use for the two-sided parallel economic search model. By identifying several unique characteristics of the equilibrium we manage to efficiently bound the strategy space that needs to be explored by the agents and propose an efficient means for extracting the distributed equilibrium strategies in common environments.