In the last few years, the memory requirements to train state-of-the-art neural networks have far exceeded the DRAM capacities of modern hardware accelerators. This has necessitated the development of efficient algorithms to train these neural networks in parallel on large-scale GPU-based clusters. Since computation is relatively inexpensive on modern GPUs, designing and implementing extremely efficient communication in these parallel training algorithms is critical for extracting the maximum performance. This paper presents AxoNN, a parallel deep learning framework that exploits asynchrony and message-driven execution to schedule neural network operations on each GPU, thereby reducing GPU idle time and maximizing hardware efficiency. By using the CPU memory as a scratch space for offloading data periodically during training, AxoNN is able to reduce GPU memory consumption by four times. This allows us to increase the number of parameters per GPU by four times, thus reducing the amount of communication and increasing performance by over 13%. When tested against large transformer models with 12-100 billion parameters on 48-384 NVIDIA Tesla V100 GPUs, AxoNN achieves a per-GPU throughput of 49.4-54.78% of theoretical peak and reduces the training time by 22-37 days (15-25% speedup) as compared to the state-of-the-art.