Many registration problems are ill-posed in homogeneous or noisy regions, and dense voxel-wise decoders can be unnecessarily high-dimensional. A sparse control-point parameterisation provides a compact, smooth deformation representation while reducing memory and improving stability. This work investigates the required control points for learning-based registration network development. We present GridReg, a learning-based registration framework that replaces dense voxel-wise decoding with displacement predictions at a sparse grid of control points. This design substantially cuts the parameter count and memory while retaining registration accuracy. Multiscale 3D encoder feature maps are flattened into a 1D token sequence with positional encoding to retain spatial context. The model then predicts a sparse gridded deformation field using a cross-attention module. We further introduce grid-adaptive training, enabling an adaptive model to operate at multiple grid sizes at inference without retraining. This work quantitatively demonstrates the benefits of using sparse grids. Using three data sets for registering prostate gland, pelvic organs and neurological structures, the results suggested a significant improvement with the usage of grid-controled displacement field. Alternatively, the superior registration performance was obtained using the proposed approach, with a similar or less computational cost, compared with existing algorithms that predict DDFs or displacements sampled on scattered key points.