Background/Objectives: The diagnosis of a rare disease, such as Sturge-Weber syndrome, in a newborn has a profound emotional impact on parents and presents a significant challenge, as they must face an unexpected and unfamiliar reality. The aim of this study was to analyze the meaning of caregiving among parents of children with Sturge-Weber Syndrome, identifying the different coping styles they adopt. Methods: A cross-sectional descriptive study was conducted with 28 parents of children with Sturge-Weber Syndrome in Spain. Data were collected using the Brief COPE inventory and the Finding Meaning Through Caregiving Scale (FMTCS). Results: Responses from 28 participants were analyzed, revealing that parents experienced a sense of fulfillment in providing meaning to the care of their children, coping with the diagnosis, and assessing their child’s development. A higher sense of care was observed in families where both parents cohabited. Additionally, as the affected descendant grew older, the perceived sense of care increased. Conclusions: The most frequently used coping strategies among parents were active coping and acceptance of the disease, regardless of sociodemographic characteristics. As parents aged, they began to adopt additional strategies such as emotional venting, positive reframing, or humor.