We study a setting of non-atomic routing in a network of m parallel links with asymmetry of information. While a central entity (such as a GPS navigation system) — a mediator hereafter — knows the cost functions associated with the links, they are unknown to the individual agents controlling the flow. The mediator gives incentive compatible recommendations to agents, trying to minimize the total travel time. Can the mediator do better than when agents minimize their travel time selfishly without coercing agents to follow his recommendations? We study the mediation ratio: the ratio between the mediated equilibrium obtained from an incentive compatible mediation protocol and the social optimum. We find that mediation protocols can reduce the efficiency loss compared to the full revelation alternative, and compared to the non mediated Nash equilibrium. In particular, in the case of two links with affine cost functions, the mediation ratio is at most 8/7, and remains strictly smaller than the price of anarchy of 4/3 for any fixed m. Yet, it approaches the price of anarchy as m grows. For general (monotone) cost functions, the mediation ratio is at most m, a significant improvement over the unbounded price of anarchy.