Embodied AI agents increasingly require parallel execution of multiple tasks, such as manipulation, conversation, and memory construction, from shared observations under distinct time constraints. Recent Mixture-of-Transformers (MoT) Vision-Language-Action Models (VLAs) architecturally support such heterogeneous outputs, yet existing inference systems fail to achieve efficient multi-task parallelism for on-device deployment because of redundant computation and resource contention. We identify isolated KV cache management as the root cause. To address this, we propose unified KV cache management, an inference design that treats the KV cache as a first-class shared resource across tasks and over time. This abstraction enables two key optimizations: cross-task KV sharing eliminates redundant prefill of shared observations, while cross-frame continuous batching decouples variable-length language decoding from fixed-rate action generation across control cycles. We implement this design for $π_{0.5}$, a popular MoT VLA, and evaluate it on both NVIDIA GeForce RTX 4090 and Jetson AGX Thor, two representative platforms for on-device VLA inference. OxyGen achieves up to 3.7$\times$ speedup over isolated execution, delivering over 200 tokens/s language throughput and 70 Hz action frequency simultaneously without degrading action quality, and we further validate the gains on a real humanoid robot with on-board Jetson AGX Thor.