Large language models (LLMs) are increasingly used for generating parallel scientific codes, with a primary focus on generating functionally correct code. Recent work has focused on generating performant code, with an emphasis on its execution time. However, energy efficiency is now recognized as a critical objective, given the significant power demands of large-scale compute systems. This paper addresses the research question of whether LLMs can generate energy-efficient parallel scientific codes when guided by empirical execution feedback. To answer this question, 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 screening generated code. We evaluate LASSI-EE using twenty-two representative scientific benchmarks and applications on NVIDIA A100 and AMD MI100 GPUs. The results indicate an average energy reduction of 36% for MI100 and 34% for A100, across trials that produced passing energy-reducing refactorings.