We present SurfDist, a convolutional neural network architecture for three-dimensional volumetric instance segmentation. SurfDist enables prediction of instances represented as closed surfaces composed of smooth parametric surface patches, specifically bicubic B\'ezier triangles. SurfDist is a modification of the popular model architecture StarDist-3D which breaks StarDist-3D's coupling of instance parameterization dimension and instance voxel resolution, and it produces predictions which may be upsampled to arbitrarily high resolutions without introduction of voxelization artifacts. For datasets with blob-shaped instances, common in biomedical imaging, SurfDist can outperform StarDist-3D with more compact instance parameterizations. We detail SurfDist's technical implementation and show one synthetic and one real-world dataset for which it outperforms StarDist-3D. These results demonstrate that interpretable instance surface models can be learned effectively alongside instance membership.