Information Extraction (IE) for semi-structured document images is often approached as a sequence tagging problem by classifying each recognized input token into one of the IOB (Inside, Outside, and Beginning) categories. However, such problem setup has two inherent limitations that (1) it cannot easily handle complex spatial relationships and (2) it is not suitable for highly structured information, which are nevertheless frequently observed in real-world document images. To tackle these issues, we first formulate the IE task as spatial dependency parsing problem that focuses on the relationship among text tokens in the documents. Under this setup, we then propose SPADE (SPAtial DEpendency parser) that models highly complex spatial relationships and an arbitrary number of information layers in the documents in an end-to-end manner. We evaluate it on various kinds of documents such as receipts, name cards, forms, and invoices, and show that it achieves a similar or better performance compared to strong baselines including BERT-based IOB taggger.