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To leverage data and computation capabilities of mobile devices, machine learning algorithms are deployed at the network edge for training artificial intelligence (AI) models, resulting in the new paradigm of edge learning. In this paper, we consider the framework of partitioned edge learning for iteratively training a large-scale model using many resource-constrained devices (called workers). To this end, in each iteration, the model is dynamically partitioned into parametric blocks, which are downloaded to worker groups for updating using data subsets. Then, the local updates are uploaded to and cascaded by the server for updating a global model. To reduce resource usage by minimizing the total learning-and-communication latency, this work focuses on the novel joint design of parameter (computation load) allocation and bandwidth allocation (for downloading and uploading). Two design approaches are adopted. First, a practical sequential approach, called partially integrated parameter-and-bandwidth allocation (PABA), yields two schemes, namely bandwidth aware parameter allocation and parameter aware bandwidth allocation. The former minimizes the load for the slowest (in computing) of worker groups, each training a same parametric block. The latter allocates the largest bandwidth to the worker being the latency bottleneck. Second, PABA are jointly optimized. Despite its being a nonconvex problem, an efficient and optimal solution algorithm is derived by intelligently nesting a bisection search and solving a convex problem. Experimental results using real data demonstrate that integrating PABA can substantially improve the performance of partitioned edge learning in terms of latency (by e.g., 46%) and accuracy (by e.g., 4%).