BALENCIAGA DIDN’T even need the kids. The brand’s spring 2018 fashion show, held last June in the verdant Bois de Boulogne park in Paris, featured a parade of models in oversize color-striped windbreakers, pale jeans similar to those that made Barack Obama dad-in-chief and bloated running shoes in the style of podiatrist-approved Asics. The actual children the male models carried were almost overkill: The 99-year-old brand was clearly celebrating that most unlikely of style icons, the dad.

Balenciaga’s father-fest was the...