This paper investigates the power allocation problem for parallel Gaussian channels from an information-theoretic perspective with the aim of maximizing the sum of mutual informations (i.e., an achievable data rate). If all the inputs are Gaussian, it is well-known that the waterfilling policy provides an optimal solution. For arbitrary input distributions, a generalization of waterfilling, so-called mercury/waterfilling, provides an optimal power allocation in terms of the minimum mean square errors (MMSEs). However, the difficulty of obtaining closed-form analytical expression of the MMSE often makes the computation of mercury/waterfilling solution challenging. This paper proposes a robust waterfilling power allocation (RPA) policy for parallel Gaussian channels when the input distributions are close to Gaussian distributions in the Kullback-Leibler divergence (relative entropy). First, it is shown that the proposed policy results in water levels that are close to the optimum in a well-defined sense. Second, tight bounds for the loss in achievable rate are given. This bounded loss property makes the proposed power allocation policy robust and approximately optimal. Both aspects are illustrated by means of different simulation setups. Finally, the RPA is argued to be scalable with the number of users on the account of the fact that it inherently uses the classical low complexity waterfilling.