Edge computing, a novel paradigm for performing computations at the network edge, holds significant relevance in the healthcare domain for extracting medical knowledge from traditional Uygur medical texts. Medical knowledge extraction methods based on edge computing deploy deep learning models on edge devices to achieve localized entity and relation extraction. This approach avoids transferring substantial sensitive data to cloud data centers, effectively safeguarding the privacy of healthcare services. However, existing relation extraction methods mainly employ a sequential pipeline approach, which classifies relations between determined entities after entity recognition. This mode faces challenges such as error propagation between tasks, insufficient consideration of dependencies between the two subtasks, and the neglect of interrelations between different relations within a sentence. To address these challenges, a joint extraction model with parameter sharing in edge computing is proposed, named CoEx-Bert. This model leverages shared parameterization between two models to jointly extract entities and relations. Specifically, CoEx-Bert employs two models, each separately sharing hidden layer parameters, and combines these two loss functions for joint backpropagation to optimize the model parameters. Additionally, it effectively resolves the issue of entity overlapping when extracting knowledge from unstructured Uygur medical texts by considering contextual relations. Finally, this model is deployed on edge devices for real-time extraction and inference of Uygur medical knowledge. Experimental results demonstrate that CoEx-Bert outperforms existing state-of-the-art methods, achieving accuracy, recall, and F1-score of 90.65%, 92.45%, and 91.54%, respectively, in the Uygur traditional medical literature dataset. These improvements represent a 6.45% increase in accuracy, a 9.45% increase in recall, and a 7.95% increase in F1-score compared to the baseline.