Jason and Sonia Green said they want to thank New Yorkers for their support following the accident.
"There are not enough words to express our gratitude to everyone who has helped our family during this difficult time," the couple said in a statement released by the hospital where their daughter Sian Green is recovering.
The 23-year-old lost a leg when she was struck by a taxi at Rockefeller Center on Tuesday. Quick-thinking bystanders including a plumber, a nurse and television personality Dr. Mehmet Oz rushed to her aid. Plumber David Justino immediately applied a tourniquet made from a belt.
Justino told The New York Times he was determined to hold on until help arrived, and then he saw Oz over his shoulder.
"I felt more comfortable when I looked up and saw his face," Justino said. "I said, 'What should I do?' and he said, 'You're doing it. You're doing it.'"
The family said it wanted especially to thank the men for their help, along with the hospital, its hotel, the airlines and the police department.
"We hope Sian will soon be on her way to recovery and will be able to personally thank all the kind people of New York City," said the couple, from Leicester, England.
Food vendor Max Crespo put Green's severed leg on ice, following the directions of a nurse whose name he didn't know.
"I took water and grabbed this bucket with ice," Crespo told the Times. "The nurse, she told me to pour water on her legs, and I did. And that is when she screamed a lot like I was pouring fire on her."
Doctors were unable to save the leg. Green's other leg was badly injured, but she likely will regain use of it with physical therapy, one of her doctors said.
The cab driver who plowed into Green said driving a taxi is too stressful and told a newspaper, "I need more suitable job." Mohammed Faysal Himon told the New York Post the accident happened after he accelerated to get around a bicyclist who pounded on his car and yelled at him.
Himon said he didn't remember much after accelerating until he saw Green's leg by his cab.
He was issued a summons, and police and prosecutors are investigating. Authorities were making moves to suspend his cab license