Multimodal perception enables robust autonomous driving but incurs unnecessary computational cost when all sensors remain active. This paper presents PRAM-R, a unified Perception-Reasoning-Action-Memory framework with LLM-Guided Modality Routing for adaptive autonomous driving. PRAM-R adopts an asynchronous dual-loop design: a fast reactive loop for perception and control, and a slow deliberative loop for reasoning-driven modality selection and memory updates. An LLM router selects and weights modalities using environmental context and sensor diagnostics, while a hierarchical memory module preserves temporal consistency and supports long-term adaptation. We conduct a two-stage evaluation: (1) synthetic stress tests for stability analysis and (2) real-world validation on the nuScenes dataset. Synthetic stress tests confirm 87.2% reduction in routing oscillations via hysteresis-based stabilization. Real-world validation on nuScenes shows 6.22% modality reduction with 20% memory recall while maintaining comparable trajectory accuracy to full-modality baselines in complex urban scenarios. Our work demonstrates that LLM-augmented architectures with hierarchical memory achieve efficient, adaptive multimodal perception in autonomous driving.