This paper is motivated by the problem of optimal flow allocation in transportation networks. We consider equilibrium and optimal flow distribution on networks of parallel links. In a parallel network, a user (or Wardrop) equilibrium is the optimal distribution of flow across alternative parallel links that minimize the effective costs of the links, while the system optimum is the optimal distribution of flow for which the average effective cost is minimal. We study Wardrop optimal networks, i.e., the networks that admit Wardrop optimal flows which satisfy both the Wardrop equilibrium and the system optimum. We first present the characterization of Wardrop optimal flows, and then propose a matrix approach for constructing Wardrop optimal networks by means of convex, strictly increasing, and continuously differentiable functions. Some special classes of Wardrop optimal networks are also introduced.