The Brazilian said Bafana Bafana, ranked 83rd in the world and previously tipped to be the worst host team yet in the 80-year history of the World Cup, had shown themselves ready for the tournament, which kicks off on Friday.
"Everyone has to respect us and we fear nobody," said Parreira as South Africa looked ahead to a last week of preparation having beaten the Danes 1-0 in Pretoria on Saturday.
"I believe the team has shown itself ready for the World Cup. We will go into the tournament with confidence and now we can start to dream about a position, about achieving something," the former World Cup winner said.
Katlego Mphela's late goal handed South Africa a third successive win and stretched to 12 games their unbeaten run since Parreira returned to coach the team in November last year.
"This was a very special win because it was against a good team with lots of experience. We played with confidence and showed we were willing to compete.
"The path we chose for the World Cup has been proven to be the right one. The players bought into the vision that they could play the ball on the ground, keep the shape and be well organised."
South Africa open the World Cup against Mexico at the cavernous Soccer City and Parreira was quick to temper any bold predictions of success in the tournament's first match.
"Mexico have proved they are a very good team. It's a very difficult group that we have and we have to take each game as a final. But the competition is open and we will go into it with confidence." They also meet Uruguay and France in their group.
Parreira also had a word of praise for a sudden surge of support for the team, reflected in capacity crowds at all their home matches over the last two months.
"The people are behind us because we have shown them the team is improving. The spectators have fallen in love with Bafana Bafana again."
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