In real-world scenarios, texts in a network are often linked by multiple semantic relations (e.g., papers in an academic network are referenced by other publications, written by the same author, or published in the same venue). Text documents and their relations form a multiplex text-rich network. Mainstream text representation learning methods use pretrained language models (PLMs) to generate one embedding for each text unit, expecting that all types of relations between texts can be captured by these single-view embeddings. However, this presumption does not hold particularly in multiplex text-rich networks. Along another line of work, multiplex graph neural networks (GNNs) directly initialize node attributes as a feature vector for node representation learning, but they cannot fully capture the semantics of the nodes' associated texts. In this paper, we propose METERN, a new framework for learning Multiplex Embeddings on TExt-Rich Networks. In contrast to existing methods, METERN uses one text encoder to model the shared knowledge across relations and leverages a small number of parameters per relation to derive relation-specific representations. This allows the encoder to effectively capture the multiplex structures in the network while also preserving parameter efficiency. We conduct experiments on nine downstream tasks in five networks from both academic and e-commerce domains, where METERN outperforms baselines significantly and consistently. The code is available at https://anonymous.4open.science/r/METER-submit-2C7B.