We analyze the syntactic sensitivity of Text-to-Speech (TTS) systems using methods inspired by psycholinguistic research. Specifically, we focus on the generation of intonational phrase boundaries, which can often be predicted by identifying syntactic boundaries within a sentence. We find that TTS systems struggle to accurately generate intonational phrase boundaries in sentences where syntactic boundaries are ambiguous (e.g., garden path sentences or sentences with attachment ambiguity). In these cases, systems need superficial cues such as commas to place boundaries at the correct positions. In contrast, for sentences with simpler syntactic structures, we find that systems do incorporate syntactic cues beyond surface markers. Finally, we finetune models on sentences without commas at the syntactic boundary positions, encouraging them to focus on more subtle linguistic cues. Our findings indicate that this leads to more distinct intonation patterns that better reflect the underlying structure.