"We played France and Mexico very well. They brought me back and to come back was great. I feel at home here. In my heart, I will remember them. What they did today was unbelievable," he said.
French coach Raymond Domenech refused to shake hands with Parreira after the final whistle and refused to discuss his actions at the post-match briefing.
Parreira said: "There was no dialogue, but I went to him as a matter of being polite as he is no longer French coach. I don't understand what happened. He said I offended the French team.
"I have never insulted them. I have always praised them."
Katlego Mphela, the man of the match who had seven shots at goal, scoring once, said he was "not sad at all" at the result. "We missed a couple of chances. The win is history in the making.
"But we gave it our all. I was unfortunate, but we fought for every ball. We beat France in the World Cup, and we try to be the biggest side in Africa again."
Bafana needed to score many goals against Les Bleus. But with French midfielder Yoann Gourcuff sent off in the 26th minute for an elbow jab at Macbeth Sibaya, a final 16 suddenly looked possible.
Two early goals, and another that was disallowed, had fans biting their nails as the other Group A decider, Mexico vs Uruguay in Rustenburg remained goalless. In the end, it was too little, too late as Uruguay beat Mexico 1-0.
But the national side went out with honour. Fans at Port Elizabeth's St Georges fan fest gave the team a standing ovation for beating the French.
At Sandton's Innisfree Park fan fest, supporters remained faithful until the last 10 minutes.
Spirits dropped, along with the temperatures, and a trickle of fans started to leave, but those who remained clapped and smiled.
Bafana's two goals set a new noise record at Cape Town's Grand Parade fan fest. The yelling and vuvuzela blasts from the 18000-strong crowd measured over 110 decibels after each goal was scored - louder than the cheer that greeted former President Nelson Mandela when he addressed the nation after his release from prison in 1990.
Dejected fans at Soweto's Elkah Stadium fan fest consoled themselves with the fact that it was Bafana's first win over France in a World Cup.
Local organising committee chief Danny Jordaan said the team won the hearts of the nation.
"Yes, they did not make it past the group stage, but they did what the country asked of them - they played with pride, passion, skill and commitment," Jordaan said.
"They gave their best. They did the national team jersey and every single South African proud and we want to thank them for what they've done."
Although the Mexicans were beaten by Uruguay, around 200 of their supporters consoled Bafana fans at the Faleno Tavern near the Royal Bafokeng stadium by singing Shosholoza and blowing their own vuvuzelas.
They said South Africans had been "very good hosts".
Jomo Sono, who took Bafana to the 2002 World Cup in Korea and Japan, said he would have liked to have seen yesterday's starting line-up throughout the tournament.
"I thought the team Parreira put out today should have been the team he started the tournament with. We played very well in the first half. Even thought France were down to 10 men, we still created a lot of chances," he said.
"But they did the country proud by winning, in front of their own fans. It's just unfortunate that we didn't qualify."
Bay United coach Khabo Zondo said: "In terms of the desired result, we didn't get there, but it was always going to be steep for us to get four in one match, when we couldn't score as many in the previous matches. But I must say there was a huge attempt."
Zondo called on the SA Football Association to start focusing on developing a team that would qualify and do well in the 2014 World Cup in Brazil, as it was "just around the corner".
Parreira said he wanted to be Bafana's "tour guide" in Brazil.
"There is a bright future for SA football. If I take anything away, I gave them a face and identity. It's a great achievement for them. They need to keep going. I want to be their guide in Brazil," he said.
Presidential spokesman Vincent Magwenya said President Jacob Zuma was "impressed" by the team's performance.
"We are already winners by just hosting the World Cup. Even though we did not go through to the second round of the tournament, we must build from what we have achieved so far.
"Yes, they did not make it past the group stage, but they did what the country asked of them - they played with pride, passion, skill and commitment. They gave their best.
"They did the national team jersey and every single South African proud, and we want to thank them for what they've done."
- Sapa-AFP quoted French captain Patrice Evra as saying France's World Cup squad would refuse all financial bonuses after their loss yesterday.