Subgrid parameterizations that represent physical processes occurring below the resolution of current climate models are an important component in producing accurate, long-term predictions for the climate. A variety of approaches have been tested to design these components, including deep learning methods. In this work, we evaluate a proof of concept illustrating a multiscale approach to this prediction problem. We train neural networks to predict subgrid forcing values on a testbed model and examine improvements in prediction accuracy which can be obtained by using additional information in both fine-to-coarse and coarse-to-fine directions.