Leveraging parallel hardware (e.g. GPUs) for deep neural network (DNN) training brings high computing performance. However, it raises data privacy concerns as GPUs lack a trusted environment to protect the data. Trusted execution environments (TEEs) have emerged as a promising solution to achieve privacy-preserving learning. Unfortunately, TEEs' limited computing power renders them not comparable to GPUs in performance. To improve the trade-off among privacy, computing performance, and model accuracy, we propose an \emph{asymmetric} model decomposition framework, \AsymML{}, to (1) accelerate training using parallel hardware; and (2) achieve a strong privacy guarantee using TEEs and differential privacy (DP) with much less accuracy compromised compared to DP-only methods. By exploiting the low-rank characteristics in training data and intermediate features, \AsymML{} asymmetrically decomposes inputs and intermediate activations into low-rank and residual parts. With the decomposed data, the target DNN model is accordingly split into a \emph{trusted} and an \emph{untrusted} part. The trusted part performs computations on low-rank data, with low compute and memory costs. The untrusted part is fed with residuals perturbed by very small noise. Privacy, computing performance, and model accuracy are well managed by respectively delegating the trusted and the untrusted part to TEEs and GPUs. We provide a formal DP guarantee that demonstrates that, for the same privacy guarantee, combining asymmetric data decomposition and DP requires much smaller noise compared to solely using DP without decomposition. This improves the privacy-utility trade-off significantly compared to using only DP methods without decomposition. Furthermore, we present a rank bound analysis showing that the low-rank structure is preserved after each layer across the entire model.