Their seven year old son was also with them but stayed behind the barrier while his parents rushed to help his brother.
"When the boy went into the quicksand, his father tried to help him and fell into the pit".
A view on Solfatara di Pozzuoli where three people died in the crater at Pozzuoli, Naples, Italy, 12 September 2017.
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Reports said the family were from Turin in northern Italy. Authorities said they believe the child fainted because of the gas fumes before falling into the crater.
It is understood the couple's older son crossed a safety barrier at the Solfatara crater at Pozzuoli and fell in the hole.
Pozzuoli's mayor, Vincenzo Figliolia, said he was in "deep pain" over the tragedy, according to La Repubblica.
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"I was at the Solfatara for work, along with other visitors we realised that something had happened and we approached the crater - still confessed the witness - I did not imagine what I would see".
Tourists are allowed to walk around the ground level Solfatara crater, which is dotted with trees, while admiring fenced off bubbling mud pools and fumeroles that spurt steam and sulphurous gases. They pulled out two bodies, then pulled us away.
Eyewitness Diego Vitagliano, a worker at the site, said: 'I saw a child run crying, I did not think I was facing the worst tragedy of my life.
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"I've been here for 40 years and such an accident has never happened", Armando Guerriero, owner of a bar located near the entrance to the volcano, told the ANSA news agency.