In this article, we present an incremental dependency parsing algorithm with an arc-eager variant of the left-corner parsing strategy. Our algorithm’s stack depth captures the center-embeddedness of the recognized dependency structure. A higher stack depth occurs only when processing deeper center-embedded sentences in which people find difficulty in comprehension. We examine whether our algorithm can capture the syntactic regularity that universally exists in languages through two kinds of experiments across treebanks of 19 languages. We first show through oracle parsing experiments that our parsing algorithm consistently requires less stack depth to recognize annotated trees relative to other algorithms across languages. This result also suggests the existence of a syntactic universal by which deeper center-embedding is a rare construction across languages, a result that has yet to be quantitatively cross-linguistically examined. We further investigate the above claim through supervised parsing experiments and show that our proposed parser is consistently less sensitive to constraints on stack depth bounds when decoding across languages, while the performance of other parsers such as the arc-eager parser is largely affected by such constraints. We thus conclude that the stack depth of our parser represents a more meaningful measure for capturing syntactic regularity in languages than those of existing parsers.