Dialogue discourse parsing is a fundamental natural language processing task. It can benefit a series of conversation-related downstream tasks including dialogue summarization and emotion recognition in conversations. However, existing parsing approaches are constrained by predefined relation types, which can impede the adaptability of the parser for downstream tasks. To this end, we propose to introduce a taskaware paradigm to improve the versatility of the parser in this paper. Moreover, to alleviate error propagation and learning bias, we design a graph-based discourse parsing model termed DialogDP. Building upon the symmetrical property of matrix-embedded parsing graphs, we have developed an innovative self-supervised mechanism that leverages both bottom-up and top-down parsing strategies. This approach allows the parsing graphs to mutually regularize and enhance each other. Empirical studies on dialogue discourse parsing datasets and a downstream task demonstrate the effectiveness and flexibility of our framework 1 .