Caring for a relative with autism spectrum disorder (ASD) entails being under chronic stress that could alter body homeostasis. Electrodermal activity (EDA) is an index of the sympathetic activity of the autonomic nervous system related to emotionality and homeostasis. This study compares EDA in response to acute stress in the laboratory between parents of people with (n = 30) and without (n = 34) ASD (caregivers and non-caregivers, respectively). Caregivers showed lower EDA in response to acute stress than non-caregivers. They also presented higher trait anxiety, anger, depression, and somatic symptoms than non-caregivers. Higher EDA was related to a worse mood and more severe somatic symptoms only in caregivers. These results could reflect an adaptive habituation to stress and establish that high EDA in response to acute stress depends on caregivers’ health.