When I decided to become a stay-at-home dad, after having earned a Ph.D. in the philosophy of education and worked in nonprofits for nearly a decade, I knew I'd learn at least as much from my kids as they would learn from me. I was excited to take part in opening up American gender roles. But I didn't anticipate how easily my anxieties about competence and failing as a caregiver would spark reactivity and self-judgment during difficult parenting...