City responded from Tuesday's disappointing home defeat to Colchester United with a battling display at Gigg Lane against a Shakers side who had won three out of their previous four home fixtures going into the match.
Although the game was evenly-fought for the most part, City arguably created the best chances in the match, with Wes Thomas and James Hanson going close in the second period.
The Bantams also saw a decent penalty appeal turned away in the first half when Hanson appeared to be nudged in the box by Bury defender Reece Brown.
Parkinson said: "I don't think you can ever underestimate the importance of a point away from home.
"Obviously you want to pick up all three when you can but the performance was definitely an improvement from Tuesday.
"For me, Hanson seems to get pushed in the box but you need those big calls to go your way.
"We had a better shape to us today (Saturday) and we had a more solid base to build our attacks from.
"It was a solid display after the disappointment of Tuesday's match and something for us to build upon.
"Bury have had some good results at home recently, they are certainly a good side here (Gigg Lane), and we don't want to undervalue the importance of this point."
Having used all three subs going into the final few minutes of the game, City finished the match with only ten men when Rory McArdle was forced off after his left shoulder temporarily popped out following an awkward fall.
Parkinson added: "There are levels of dislocation you can suffer with a shoulder and we just hope Rory's wasn't too serious.
"He and Reece (Burke) were excellent against two experienced attackers in Leon Clarke and Ryan Lowe so he will be missed. We do have players to come in like Christopher Routis and Nathan Clarke if needed though.
"Mezza (James Meredith) had a bit of sickness and diarrhoea this morning but there was never any doubt that he wouldn't put himself up to play, and I was pleased with his contribution as well."
While City were drawing with Bury, other results around the grounds in the Sky Bet League 1 means Parkinson's side now sit three points off the Play-Off places with a game in hand on Millwall and Barnsley in fifth and sixth respectively.
The Bantams now look ahead to their rearranged game with league leaders Burton Albion on Tuesday at the Coral Windows Stadium.
Parkinson continued: "You look at some of the results again today, it just shows you how unpredictable this division can be. This point at Bury could prove to be an important one for us.
"Hopefully we can get a good crowd in right behind the lads for Tuesday night (vs Burton) and we can make it an evening where we really make home advantage count for a lot."