Large language model (LLM) inference workloads have emerged as a critical reliability challenge for cloud GPU systems. Unlike traditional workloads, the highly structured execution of LLMs creates large power oscillations. These oscillations become a vulnerability when their frequency aligns with the resonant modes of a GPU's power delivery network (PDN), leading to excessive voltage droops and unreliable operation. In this work, we present the first comprehensive profiling of LLM-induced power oscillations, revealing that many workloads generate oscillatory patterns in the MHz range-critically aligning with typical GPU PDN resonant frequencies and leading to excessive voltage droops. To systematically investigate this phenomenon, we developed a novel stressmark framework that generates workloads with controllable, high-frequency power oscillations and voltage droops. Our evaluation shows that operating at a resonant frequency induces voltage droops up to$2 \times$larger than conventional workloads, exceeding critical noise margins. Critically, we find that real LLM workloads operating even near these frequencies generate significant voltage droops greater than 100 mV. Based on these findings, we propose a kernel staggering technique that mitigates this threat by shifting power oscillation frequencies away from resonance frequency, successfully reducing voltage droops and reducing reliability concerns. This work provides the first systematic understanding of LLM-PDN resonance and offers a practical solution to improve GPU reliability in AI cloud environments.