Scaling CNN training is necessary to keep up with growing datasets and reduce training time. We also see an emerging need to handle datasets with very large samples, where memory requirements for training are large. Existing training frameworks use a data-parallel approach that partitions samples within a mini-batch, but limits to scaling the minibatch size and memory consumption makes this untenable for large samples. We describe and implement new approaches to convolution, which parallelize using spatial decomposition or a combination of sample and spatial decomposition. This introduces many performance knobs for a network, so we develop a performance model for CNNs and present a method for using it to automatically determine efficient parallelization strategies. We evaluate our algorithms with microbenchmarks and image classification with ResNet-50. Our algorithms allow us to prototype a model for a mesh-tangling dataset, where sample sizes are very large. We show that our parallelization achieves excellent strong and weak scaling and enables training for previously unreachable datasets.