Graph neural networks (GNNs) have been proven effective in the fast and accurate prediction of nuclear magnetic resonance (NMR) chemical shifts of a molecule. Existing methods, despite their effectiveness, suffer from high space complexity and are therefore limited to relatively small molecules. In this work, we propose a scalable GNN for NMR chemical shift prediction. To reduce the space complexity, we sparsify the graph representation of a molecule by regarding only heavy atoms as nodes and their chemical bonds as edges. To better learn from the sparsified graph representation, we improve the message passing and readout functions in the GNN. For the message passing function, we adapt the attention mechanism and residual connection to better capture local information around each node. For the readout function, we use both node-level and graph-level embeddings as the local and global information to better predict node-level chemical shifts. Through the experimental investigation using 13C and 1H NMR datasets, we demonstrate that the proposed method yields higher prediction accuracy and is more scalable to large molecules having many heavy atoms.