The 31-year-old midfielder, who spent most of the evening scrapping for possession in a combative game controlled by the French, echoed new manager Roy Hodgson's pragmatic view that a draw was a welcome outcome.
"I don't know if you can win tournaments playing like that but it's a good foundation for us," Parker told reporters. "We all put in a good shift and we got a good result. France didn't have any clear chances and we'll take the 1-1."
France stretched their unbeaten run to 22 games and had 65 per cent of the possession but were left the more frustrated team to Hodgson's obvious pleasure.
"We played well against a top team that ask questions when coming up against you," Hodgson said. "They don't panic in the final third so we had to be disciplined. I have to be happy with the result.
"We kept them at bay and there were not many situations where I was panicking, but if we had more luck with the final ball we could have scored a second goal."
Hodgson, who gave 18-year-old Alex Oxlade-Chamberlain only his second international start, praised the Arsenal forward.
"Oxlade-Chamberlain did quite well, he was dangerous," he said. "He has to be happy with his performance but it was a great team performance. Everyone did their best to keep us in the game.
"France are one of the favourites in the group, but I have lots of respect for Sweden and Ukraine.
"The final ball and decision making needs to improve. Sometimes we crossed too early, or had one touch passes when not required, but it is a good sign that everyone wants to do well."
As expected, England were durable but lacked attacking flair, not that captain Steven Gerrard was too concerned after seeing Samir Nasri strike a fine equaliser after Joleon Lescott had nodded England ahead.
"We would have been delighted with a win, but as you saw tonight France have got fantastic players." Gerrard said.
"We caused them as many problems as they caused us so we're satisfied with the performance."
Goalkeeper Joe Hart was beaten at his near post by Nasri's crisp strike.
"Samir's a clever player," he said of his Manchester City team mate. "We dropped off a little bit. It was a good goal for them, but we have to concentrate on our good performance.
"You can question the team all you want, but you'll always get belief, pride and quality. People forget that we're England and we're a good side - and in that first 25 minutes they weren't ready for us."
