Another child, aged seven, survived the tragedy.
According to an initial reconstruction by the Campania Civil Protection Department the victims fell into a pit of boiling mud.
The three-metre-deep pit opened up as the mother and father followed their son, who is thought to have entered an area forbidden to visitors, and the bodies were later recovered by firefighters.
Other visitors were told to exit the site and it was closed after the incident occurred.
The surviving child was temporarily placed into the care of local social workers and a psychologist, until family members arrived at the scene.
The victims have been identified as Tiziana Zaramella, a 42-year-old woman originally from the province of Turin, her partner Massimiliano Carrer, 45, and 11-year-old Lorenzo.
The family lived in Meolo, a town in the province of Venice, and Ms Zaramella worked at Venice Marco Polo Airport.
“I'm distraught by what happened at the Solfatara volcano,” Pozzuoli Mayor Vincenzo Figliolia said following the discovery.
“I feel deep pain for this inhuman tragedy involving three tourists. I express the full support of the community of Pozzuoli to the families of the victims.”