Individuals with autism spectrum disorder (ASD) exhibit unique speech patterns that challenge conventional automatic speech recognition (ASR) systems. However, research on ASD-adapted ASR models remains limited. This study explores fine-tuning strategies for ASD-specific ASR models using Whisper, comparing full fine-tuning, selective fine-tuning, adapter tuning, and LoRA-based fine-tuning. Experiments using a small-scale Korean ASD speech dataset demonstrate that adapter tuning and LoRA significantly reduce the character error rate (CER) while reducing trainable parameters. In case of Whisper-small, adapter tuning and LoRA improve the CER by 7.22% and 10.14% over full fine-tuning, respectively. Furthermore, LoRA improved CER by 10.35% and 10.15% with Whisper-base and Whisper-large-v2 models compared to full fine-tuning. These results demonstrate the adaptation efficiency and effectiveness of LoRA for low resource ASD speech dataset.