Learned image compression (LIC) holds great potential for advancing various computer vision applications. However, traditional fixed-rate LIC methods require training separate models for each target bitrate, resulting in high computational costs and limited scalability. To address these challenges, we propose a parameter-efficient variable-rate compression framework that combines BitFit and Low-Rank Adaptation (LoRA). Specifically, we fine-tune only the bias terms and apply LoRA to the fully connected layers within the transformer blocks, while keeping all other parameters frozen. A pretrained or newly trained fixed-rate model serves as the backbone, which is adapted to other target rates through lightweight fine-tuning. Furthermore, we introduce trainable channel-wise quantization step sizes to enhance compression performance across different bitrates. Experimental results demonstrate that our method achieves competitive rate-distortion performance with significantly reduced training time and memory consumption, offering a practical solution for scalable variable-rate image compression.