In this paper, we consider a distributionally robust resource planning model inspired by a real-world service industry problem. In this problem, there is a mixture of known demand and uncertain future demand. Prior to having full knowledge of the demand, we must decide upon how many jobs we plan to complete on each day in the planning horizon. Any jobs that are not completed by the end of their due date incur a cost and become due the following day. We present two distributionally robust optimisation (DRO) models for this problem. The first is a non-parametric model with a phi-divergence based ambiguity set. The second is a parametric model, where we treat the number of uncertain jobs due on each day as a binomial random variable with an unknown success probability. We reformulate the parametric model as a mixed integer program and find that it scales poorly with the sizes of the ambiguity and uncertainty sets. Hence, we make use of theoretical properties of the binomial distribution to derive fast heuristics based on dimension reduction. One is based on cutting surface algorithms commonly seen in the DRO literature. The other operates on a small subset of the uncertainty set for the future demand. We perform extensive computational experiments to establish the performance of our algorithms. We compare decisions from the parametric and non-parametric models, to assess the benefit of including the binomial information.