Some text generation tasks, such as Attribute Value Extraction (AVE), require decoding multiple independent sequences from the same document context. While standard autoregressive decoding is slow due to its sequential nature, the independence between output sequences offers an opportunity for parallelism. We present Hyper-Parallel Decoding, a novel decoding algorithm that accelerates offline decoding by leveraging both shared memory and computation across batches. HPD enables out-of-order token generation through position ID manipulation, significantly improving efficiency. Experiments on AVE show that attribute-value pairs are conditionally independent, enabling us to parallelize value generation within each prompt. By further stacking multiple documents within a single prompt, we can decode in parallel up to 96 tokens per prompt. HPD works with all LLMs, and reduces both inference costs and total inference time by up to 13.8X without compromising output quality, potentially saving hundreds of thousands of dollars on industry AVE tasks. Although designed for attribute extraction, HPD makes no assumptions unique to the AVE domain and can in theory be applied to other scenarios with independent output structures.