Automatically extracting interpersonal relationships of conversation interlocutors can enrich personal knowledge bases to enhance personalized search, recommenders and chatbots. To infer speakers' relationships from dialogues we propose PRIDE, a neural multi-label classifier, based on BERT and Transformer for creating a conversation representation. PRIDE utilizes dialogue structure and augments it with external knowledge about speaker features and conversation style.Unlike prior works, we address multi-label prediction of fine-grained relationships. We release large-scale datasets, based on screenplays of movies and TV shows, with directed relationships of conversation participants. Extensive experiments on both datasets show superior performance of PRIDE compared to the state-of-the-art baselines.