Training large language models (LLMs) is fundamentally constrained by limited device memory and costly inter-device communication. Although pipeline parallelism alleviates memory pressure by partitioning models across devices, it incurs activation communication overhead that scales linearly with sequence length, limiting efficiency in long-context training. Recent weight-passing approaches (e.g., WeiPipe) mitigate this by transmitting model weights instead of activations, but suffer from redundant peer-to-peer (P2P) transfers and underutilized intra-node bandwidth. We propose TawPipe--topology-aware weight pipeline parallelism, which exploits hierarchical bandwidth in distributed clusters for improved communication efficiency. TawPipe: (i) groups devices based on topology to optimize intra-node collective and inter-node P2P communication; (ii) assigns each device a fixed shard of model weights and gradients, avoiding redundant transfers; and (iii) overlaps communication with computation to hide latency. Unlike global collective operations used in fully sharded data parallelism (FSDP), TawPipe confines most communication within node boundaries, significantly reducing cross-node traffic. Extensive experiments on up to 24 GPUs with LLaMA-style models show that TawPipe achieves superior throughput and scalability compared to state-of-the-art baselines.