A win would have seen Bournemouth put clear daylight between them and their automatic promotion rivals.
But the Cherries found it hard to break down a determined Boro back-line, managing only two shots on target in the 90 minutes.
Head coach Parker said: “I think that’s a big point, that’s how I see it.
“Of course we wanted all three points but Boro are a well-organised team and we knew they would be hard to play against.
“We had a couple of little half chances and Jefferson hit the bar and did not follow the ball in.
“Those are the fine margins we are working with but every game we play is another one ticked off. I think that’s a point gained.
“I was pleased with the second half and saw a group of men trying to win the game.
“They limited us to very few opportunities and I always felt the game would be tight.
“We are in a fantastic place with six games to go and we would not want it any other way.
“I understand the anxiousness and the desperation of everyone wanting to get this season done and dusted and for us to get promoted.
“We need to stay focused and very calm. We will be alright.”
Boro striker Duncan Watmore came closest to breaking the deadlock in a drab first half with a fierce long-range shot well saved by Mark Travers.
Jefferson Lerma headed against the underside of the crossbar in the second half as Bournemouth pushed for a first win in three games.
Boro could have had a 26th-minute penalty when Adam Smith clearly handled an Isaiah Jones cross.
But visiting boss Chris Wilder kept his complaints to a minimum, saying: “We haven’t been given it so it’s ifs, buts and maybes. I don’t think there was much in the game today at all.
“We’ve had a couple of chances, they’ve had a couple of chances.
“I think that epitomizes the game and maybe how the games are going to go from now until the end of the season.
“We’re delighted with the point.
“I believe that they will succeed in getting into the Premier League and for us to haul ourselves in and around the top six, which we are.
“That’s the aim. There’s a lot of football between now and the end of the season. I haven’t got a crystal ball – who knows what it’s going to take?
“We’ve played the two best teams in the division, Fulham and Bournemouth, in the space of 10 days. They are both really good sides and the performance levels have been good.”