In this paper, we present a neural model for joint dropped pronoun recovery (DPR) and conversational discourse parsing (CDP) in Chinese conversational speech. We show that DPR and CDP are closely related, and a joint model benefits both tasks. We refer to our model as DiscProReco, and it first encodes the tokens in each utterance in a conversation with a directed Graph Convolutional Network (GCN). The token states for an utterance are then aggregated to produce a single state for each utterance. The utterance states are then fed into a biaffine classifier to construct a conversational discourse graph. A second (multi-relational) GCN is then applied to the utterance states to produce a discourse relation-augmented representation for the utterances, which are then fused together with token states in each utterance as input to a dropped pronoun recovery layer. The joint model is trained and evaluated on a new Structure Parsing-enhanced Dropped Pronoun Recovery (SPDPR) dataset that we annotated with both two types of information. Experimental results on the SPDPR dataset and other benchmarks show that DiscProReco significantly outperforms the state-of-the-art baselines of both tasks.