Large multimodal models (LMMs) typically employ an encoding module to transform multimodal data inputs into embeddings, which are then fed to language models for further processing. However, efficiently serving LMMs remains highly challenging due to the inherent complexity of their inference pipelines. Traditional serving engines co-locate the encoding module and the language model, leading to significant resource interference and tight data dependency. Recent studies have alleviated this issue by disaggregating the encoding module from the model, following a design style of prefill-decode disaggregation. Nevertheless, these approaches fail to fully exploit parallelism both within individual requests (intra-request) and across multiple requests (inter-request). To overcome the limitation, we propose REDServe, an LMM inference system that efficiently orchestrates intra- and inter-request pipelines. REDServe is designed to reduce low latency and maximize parallelism at both intra- and inter-request granularities. Built on the disaggregated architecture of the encoding module and language model, REDServe adopts a fine-grained scheduling method that overlaps multimodal encoding with the forward computation of the language model within a single request. For inter-request pipeline, REDServe leverages schedulable tokens and token budgets to balance computational loads across micro-batches. Combined with chunked prefill, this enables a novel scheduling strategy that coordinates the execution of intra- and inter-request pipelines. Experimental evaluations on representative LMMs show that REDServe achieves substantial latency reduction of up to 66% while improving throughput by up to 109%, significantly outperforming existing serving approaches.