While large language models (LLMs) are increasingly used for generating parallel scientific codes, most efforts emphasize functional correctness, often overlooking performance, especially energy efficiency. We propose LASSI-EE, an automated LLM-based refactoring framework that generates energy-efficient parallel codes through a multi-stage, iterative approach integrating runtime power profiling, energy-aware prompting, self-correcting feedback loops, and an LLM-as-a-Judge agent for automated screening of code solutions. We introduce energy-reduction@k, a novel metric that quantifies expected energy reduction when generating k code candidates and selecting the most energy-efficient, enabling systematic evaluation of multi-attempt generation strategies. Evaluating 20 HeCBench applications and two miniApps on NVIDIA A100 and AMD MI100 GPUs, a single run (k=1) with LASSI-EE delivers refactored parallel codes with an average 29% expected energy reduction at an 81% pass rate, representing a 2.8x improvement over vanilla LLM prompting. Multiple runs (k=3) achieve an average 48% expected energy reduction at a 97% pass rate. These results are consistent across devices, demonstrating LASSI-EE's effectiveness across diverse hardware architectures.