Cherries striker Dominic Solanke missed two late chances to win it for the visitors, but the club's top scorer shot and headed over - the latter a good chance from a 90th-minute corner - as Peterborough held on for a precious point.
The result leaves the visitors behind second place West Brom on goal difference after the Baggies' 4-0 win at Cardiff a day earlier, while an eighth home point of the season (compared to none away) was not enough for Posh to move out of the relegation places.
Parker said: "It was a bit passive. We lacked intent and played the game in front of them.
"To their credit Peterborough are an organised team and limited the space.
"There was a reaction in the second half and we had an attacking threat. We've just not managed to put the chances away. We gave ourselves 45 minutes to win a game of football and that's not us.
"We've been very clinical and very fluid (previously) but we've not put the ball away. We had some good chances but looked a bit off it. That's football. They're humans and sometimes have those moments.
"The second half is where we need to stay.
"Dom's been fantastic since I've come here and missed some chances, but he puts himself in good areas for the team. It was just not his night."
Solanke's late efforts aside, this was a match of few chances.
The striker and team-mate Morgan Rogers both had efforts saved by Peterborough goalkeeper Dai Cornell, while at the other end Jorge Grant was unable to convert Harrison Burrows' right-wing cross on the slide under pressure from former England defender Gary Cahill.
Delighted Peterborough manager Darren Ferguson said: "I spoke to the players about how good our home form has been.
"The players got the game plan right to a man. We played as a team against a side who are ahead of most others in the league. We've not had many chances in the match.
"The fans are used to us bombing forward and attacking, but you can't do that against Bournemouth as they would have cut us to shreds. The atmosphere was lacking because of that, but we were very solid and did what we worked on.
"We've had eight years in League One and they've had five years in the Premier League, so I'm really delighted. Some of the football was excellent in the first half, and the subs did well when they came on.
"I've got two weeks to sort out our away form as we are a completely different team at home."