Recent studies on parameter-efficient fine-tuning (PEFT) have introduced effective and efficient methods for fine-tuning large language models (LLMs) on downstream tasks using fewer parameters than required by full fine-tuning. Low-rank decomposition adaptation (LoRA) significantly reduces the parameter count to 0.03% of that in full fine-tuning, maintaining satisfactory performance when training only two low-rank parameters. However, limitations remain due to the lack of task-specific parameters involved in training. To mitigate these issues, we propose the Lottery Rank-Pruning Adaptation (LoRPA) method, which utilizes the Lottery Ticket Hypothesis to prune less significant parameters based on their magnitudes following initial training. Initially, LoRPA trains with a relatively large rank size and then applies pruning to enhance performance in subsequent training with fewer parameters. We conducted experiments to compare LoRPA with LoRA baselines, including a setting with a relatively large rank size. Experimental results on the GLUE dataset with RoBERTa demonstrate that LoRPA achieves comparable results on the base scale while outperforming LoRA with various rank sizes by 0.04% to 0.74% on a large scale across multiple tasks. Additionally, on generative summarization tasks using BART-base on the CNN/DailyMail and XSum datasets, LoRPA outperformed LoRA at the standard rank size and other PEFT methods in most of the metrics. These results validate the efficacy of lottery pruning for LoRA in downstream natural-language understanding and generation tasks.