Extensive research has shown elevated mental health difficulties among parents of children with autism compared to other parents. Although several studies have explored factors related to mental health among parents of children with autism, the factors that influence and promote well-being remain poorly characterised. Parents of young, newly diagnosed autistic children may also be particularly vulnerable to stressors that impact mental health and well-being. We examined child-, parent-, and family/socioeconomic factors associated with concurrent mental health and well-being among 136 parents of young children with autism, aged 13-48 months. Parental mental health was predicted by both trait negative emotionality and reported child autism symptoms, while well-being was predicted by parent factors alone, including trait extraversion and mindfulness. Broader child characteristics and family/socioeconomic contextual factors made no significant contribution in regression models. While the mental health and well-being of parents with young autistic children are associated with one another, unique predictors seem to exist. That well-being was uniquely predicted by a modifiable parent characteristic -- mindfulness -- suggests the potential for early supports to bring direct benefits for parents, in the context of raising a young child with autism.