One of our first infractions at Ghibli Park was hoisting our 1-year-old onto the polyester tummy of a woodland spirit creature. Another was letting him slip under a barricade and shelter inside a furry bus with cat eyes for headlights.

“He’s not following the protocol,” I told my wife, as the staff overseeing the cat-bus play zone looked on anxiously.

“He’s making a mockery of it,” she said. But we didn’t stop him.

Ghibli Park, which opened in November outside Nagoya, Japan, pays homage to the eccentric, enchanting films of Studio Ghibli, a company co-founded in the 1980s by the director Hayao Miyazaki. We took our two toddlers there because their favorite movie is “My Neighbor Totoro,” a beloved 1988 Miyazaki film starring the spirit creature and its cat-bus sidekick.

As parents, we thought it would be fun for our boys, 3 and 1, to experience a “Totoro” immersion. And as longtime Ghibli fans, we were keen to see what the place looked like.

American visitors may…