Traditional NLP has long held (supervised) syntactic parsing necessary for successful higher-level semantic language understanding (LU).The recent advent of end-to-end neural models, self-supervised via language modeling (LM), and their success on a wide range of LU tasks, however, questions this belief.In this work, we empirically investigate the usefulness of supervised parsing for semantic LU in the context of LM-pretrained transformer networks.Relying on the established fine-tuning paradigm, we first couple a pretrained transformer with a biaffine parsing head, aiming to infuse explicit syntactic knowledge from Universal Dependencies treebanks into the transformer.We then fine-tune the model for LU tasks and measure the effect of the intermediate parsing training (IPT) on downstream LU task performance.Results from both monolingual English and zero-shot language transfer experiments (with intermediate target-language parsing) show that explicit formalized syntax, injected into transformers through IPT, has very limited and inconsistent effect on downstream LU performance.Our results, coupled with our analysis of transformers' representation spaces before and after intermediate parsing, make a significant step towards providing answers to an essential question: how (un)availing is supervised parsing for high-level semantic natural language understanding in the era of large neural models?