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This paper is concerned with deriving an optimal flow and routing policy for two-node parallel link communication networks with multiple competing users. The model assumes that every user has a flow demand which is not fixed and which needs to be optimally routed over the network links. The flow and routing policy for each user is derived by simultaneously maximizing the total throughput and minimizing the expected delay for that user. Instead of considering the utility functions which combine the two objectives in a multiplicative fashion, as is typically done in the literature, We consider the utility functions that combine them in a linear additive fashion. We introduce two preference constants into each utility function so that each user can adjust its utility to reflect its own preferences. Because of the fact that the network resources are shared in a competitive manner by all users, this multiuser multi-objective optimization problem is formulated as a non-cooperative game problem among all the users. When the preference constants satisfy a condition, we show that this network game admits a non-symmetric flow and routing control policy that satisfies the Nash equilibrium solution. We discuss the properties of this equilibrium and illustrate the results with an example.