The AI inference industry keeps models loaded in GPU memory around the clock to avoid cold-start latency, implicitly treating idle power as a fixed cost of readiness. Yet the structure of this cost has never been empirically decomposed - and never across GPU architectures. We present the first cross-architecture measurement of idle GPU power as a function of VRAM allocation, combining 18 days of production telemetry (335,267 samples, 14 H100 GPUs) with controlled dose-response experiments on three GPU architectures spanning three memory technologies: NVIDIA H100 (HBM3, 80 GB), A100 (HBM2e, 80 GB), and L40S (GDDR6, 48 GB). We observe that idle power is piecewise constant on all three architectures: the CUDA context forces a discrete DVFS transition consuming +26-66 W over bare idle (26-50 W on HBM architectures, 66 W on GDDR6), while the marginal VRAM effect is bounded below measurement relevance ($|β| < 0.02$ W/GB) on every device tested. The CUDA context accounts for >98% of the parking tax regardless of memory technology. We validate this finding with a real HuggingFace model (Qwen2.5-7B) on all three architectures, confirming <0.5 W difference from empty tensors on every device, and capture cold-start power profiles during model loading. We derive a cold-start breakeven model showing energy-optimal behavior depends on request arrival rate and loading latency - not model size - with breakeven intervals of 1-5 minutes. Our results identify a constraint consistent across all tested architectures: idle-with-context power is determined by DVFS state, not memory occupancy.