Parsing spoken dialogue presents challenges that parsing text does not, including a lack of clear sentence boundaries. We know from previous work that prosody helps in parsing single sentences (Tran et al. 2018), but we want to show the effect of prosody on parsing speech that isn't segmented into sentences. In experiments on the English Switchboard corpus, we find prosody helps our model both with parsing and with accurately identifying sentence boundaries. However, we find that the best-performing parser is not necessarily the parser that produces the best sentence segmentation performance. We suggest that the best parses instead come from modelling sentence boundaries jointly with other constituent boundaries.