The "Palace of Fun", which left Civitavecchia at about 19:00, ran aground at Giglio island at about 21:45.
Laurence Davis, 60, celebrated his birthday on December 30 and his wife Andy's is on January 18.
They'd decided on the boat trip to celebrate their birthdays and were having dinner with friends when disaster struck.
“My dad didn't hear anything, but felt something was wrong. Then plates and cutlery started sliding around. Chaos erupted," Bradley said.
Panic-stricken passengers tried to get to lifeboats. Bradley said when his parents couldn't get a lifeboat on the one deck, they ran to another.
"There was pandemonium. The ship started listing more and more and someone fell against my father. He was pinned against a wall."
Laurence managed to free himself. It was then that he realised that they had to get off the ship.
"My dad put his glasses in his jacket pocket. Thank heavens, because he can't see without them.
"Then they jumped four storeys into the cold sea below," Bradley said.
They swam about 100m to a small island. The rocks were sharp and the couple took about 45 minutes to get to the top.
Their only light was a small torch on Laurence's life jacket.
There were already 6 people on the island.
The worst for Laurence was the fact that there were about 100 others trying to get to the island, Bradley said.
They were later picked by a ferry and taken to an island.
"It looked like a war zone with ambulances and people searching for loved ones. It broke my dad's heart.
Bradley said he'd only managed to get hold of his father on Saturday night. "I only believed that he was alive after speaking to him."
Bradley said his dad and Andy are very traumatised.
Thought it was an act
Another South African, Mike van Dijk and his wife were at a magician show when the dustbins started falling over.
"I thought it was part of the act, because all of a sudden the curtains started blowing in one direction.
But when he saw the stage manager's face, he realised it was not part of the act and they had to get out of there.
He said there was chaos as people tried to get to life rafts.
Helicopters and ferries were used to rescue passengers - many of them with fractures and hypothermia.
A spokesperson for the department of international affairs, Nelson Kgwete, confirmed that the Van Dijk and Davis families were safe and in a hotel in Rome.