The family of four - including a seven-year-old boy who survived - were visiting Solfatara, one of the main attractions at the popular Campi Flegrei "Burning Fields" volcanic tourist area, about 30 kilometres east of Naples.
"We are still trying to understand what happened," a Campi Flegrei park official told AFP by telephone.
A view on Solfatara di Pozzuoli where three people died in the crater at Pozzuoli, Naples, Italy, 12 September 2017.
According to media reports, the 11-year-old boy was the first to fall after crossing the barriers.
His father, then his mother, both fell in as they tried to rescue him. Authorities believe they died from inhaling gas fumes.
Only the seven-year-old stayed behind the barriers and was found suffering from shock.
The Campi Flegrei is an area with more than 30 volcanic craters, whose possible eruptions worry scientists.