On a recent Sunday, Silvia Rezzonico pushed a stroller through the glorious gardens of Villa Belgiojoso, winding past parents chasing their toddlers and families lounging on blankets.

Under a black walnut tree, she paused to look down at the baby boy dozing angelically in the pram.

“He’s not really mine,” she said.

But he was her coveted ticket into the park.

The secret gardens of Villa Belgiojoso, entered through an anonymous little door, are off limits to adults unaccompanied by a minor. A few blocks from Milan’s renowned fashion district and just a 10-minute walk from the frenzied Porta Venezia neighborhood, the gardens look like a scene out of a Milanese “Mary Poppins.” They are limited, free of charge, to visitors under the age of 13 and their guardians.

That’s why Ms. Rezzonico, a 38-year-old speech therapist with no children of her own, had asked a friend if she could latch herself onto his baby.