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Syntax — the study of the hierarchical structure of language — has long featured as a prominent research topic in the field of natural language processing (NLP). Traditionally, its role in NLP was confined towards developing parsers: supervised algorithms tasked with predicting the structure of utterances (often for use in downstream applications). More recently, however, syntax (and syntactic theory) has factored much less into the development of NLP models, and much more into their analysis. This has been particularly true with the nascent relevance of language models: semi-supervised algorithms trained to predict (or infill) strings given a provided context. In this dissertation, I describe four separate studies that seek to explore the interplay between syntactic parsers and language models upon the backdrop of dependency syntax. In the first study, I investigate the error profiles of neural transition-based and graph-based dependency parsers, showing that they are effectively homogenized when leveraging representations from pre-trained language models. Following this, I report the results of two additional studies which show that dependency tree structure can be partially decoded from the internal components of neural language models — specifically, hidden state representations and self-attention distributions. I then expand on these findings by exploring a set of additional results, which serve to highlight the influence of experimental factors, such as the choice of annotation framework or learning objective, in decoding syntactic structure from model components. In the final study, I describe efforts to quantify the overall learnability of a large set of multilingual dependency treebanks — the data upon which the previous experiments were based — and how it may be affected by factors such as annotation quality or tokenization decisions. Finally, I conclude the thesis with a conceptual analysis that relates the aforementioned studies to a broader body of work concerning the syntactic knowledge of language models.