Tabular data is critical across diverse domains, yet high-quality datasets remain scarce due to privacy concerns and the cost of collection. Contemporary approaches adopt large language models (LLMs) for tabular augmentation, but exhibit two major limitations: (1) dense dependency modeling among tabular features that can introduce bias, and (2) high computational overhead in sampling. To address these issues, we propose SPADA for SPArse Dependency-driven Augmentation, a lightweight generative framework that explicitly captures sparse dependencies via an LLM-induced graph. We treat each feature as a node and synthesize values by traversing the graph, conditioning each feature solely on its parent nodes. We explore two synthesis strategies: a non-parametric method using Gaussian kernel density estimation, and a conditional normalizing flow model that learns invertible mappings for conditional density estimation. Experiments on four datasets show that SPADA reduces constraint violations by 4% compared to diffusion-based methods and accelerates generation by nearly 9,500× over LLM-based baselines.