In this paper, we tackle the Aerial Vision-and-Dialog Navigation (AVDN) task in the training-free setting for resource-efficient high-altitude UAV navigation.Naively applying MLLMs leads to unreliable navigation due to weak directional grounding and the lack of explicit spatial memory.To address these issues, we propose PSC-AVDN, a training-free framework that tightly couples a three-stage Parsing-Search-Confirmation reasoning pipeline with a Structured Spatial Memory (SSM).The parsing stage uses an LLM to convert ambiguous dialogue instructions into stable geometric directional and destination cues.A Search Chain-of-Thought (S-CoT) then performs stepwise target exploration under high-altitude observations, and a Confirmation Chain-of-Thought (C-CoT) conducts fine-grained verification around candidate regions to resolve visual ambiguity.Meanwhile, SSM integrates three complementary sources of spatial cues, including multi-scale visual observation, spatial visual memory, and structured geometric memory to provide global spatial context and long-horizon consistency.Extensive experiments on ANDH and ANDH-Full show that PSC-AVDN establishes new state-of-the-art performance in the training-free setting, matching or surpassing several finetuned methods.Code will be publicly available at: https://github.com/QY6616/PSC-AVDN