The paradigm shift in the field of Deep Learning, due to the emergence of Foundation Models and, in particular, Large Language Models has transformed how models operate. These models handle multiple tasks by leveraging their inherent transfer learning capabilities. While powerful, these models raise concerns about computational and environmental costs. In response, we demonstrate that our lightweight methodology for adapting pre-trained Large Language Models to Time Series Forecasting using Parameter-Efficient Fine-Tuning techniques such as Low-Rank Adaptations, called LLIAM, is capable of significantly reducing total training time and resource consumption while maintaining accuracy comparable to other state-of-the-art Deep Learning models. This work contributes to the advancement of Green AI by demonstrating a more sustainable and accessible utilization of these large models for specialized tasks.