Father of two and "dad influencer" Diego Di Franco takes his daughter Eva to a birthday party, in Trezzano sul Naviglio, Italy, on April 9. | REUTERS

ROME –

Every afternoon in a Milan suburb, Diego Di Franco picks up his children from school, manages after-school activities and prepares dinner, tasks traditionally associated with Italian mothers.

The routine is unremarkable in Italy, except that he is a father and he shares it online. Italy's parliament in February rejected a proposal to equalize maternity and paternity leave, but Di Franco and a growing number of what are being called "dad influencers" are reshaping how fatherhood looks in a country still struggling ​to reconcile work, family and gender equality.…