City spent virtually all of the second half down to ten men following Stephen Darby's dismissal less than twenty seconds after the restart.
Despite the numerical disadvantage and finding themselves a goal down midway through the second period though, Parkinson's men wrestled some of the initiative back in the game later in the half.
Alan Sheehan forced a fine save out of United 'keeper Mark Howard with a fierce free-kick from just outside the box, while Jason Kennedy went agonisingly close when he headed against the post from close range.
Around a minute later after Kennedy's near-miss, however, a loose touch from Andrew Davies allowed Blades sub Marc McNulty to race clear and wrap up the game for the visitors with a clinical one-on-one finish.
Parkinson said: "The first half was tough - there wasn't too many goalscoring opportunities for us. We knew they (Sheffield United) would go with a 4-5-1 and sit everyone behind the ball when they weren't in possession.
"There were times when the ball went up to Hans (James Hanson) and we could have been a bit sharper around him. There needed to be a sharper anticipation from us in and around the box.
"We worked hard and contained them after Stephen's (Darby) dismissal and it took a quality finish from them to get one-nil up.
"We changed it after that and put Yeatesy (Mark Yeates) just in behind Jon Stead and we've then had some decent opportunities to score. The best one came with Jason Kennedy's header, which has hit the post.
"He'll feel he should have scored from that. We've had a couple of chances from set-piece situations as well - we have been a bit unfortunate really.
"I didn't feel they were threatening us too much in the second period, despite the man advantage.
"We couldn't quite get that equaliser when we had the chance but I can't fault the lads' efforts tonight (Saturday). They gave us everything.
"The second goal is something we can't feel legislate for because the ball has bounced over Andrew Davies' foot. The lad (Marc McNulty) has stuck it away and it was game over really after that.
"We had real momentum in our favour around the time of Jason's chance and the crowd were really with us. When you are two-nil down with ten-man, although we kept plugging away, but it was always going to be tough.
"It just wasn't to be for us tonight. We've had one tough decision go against us and one moment of misfortune with Andrew (Davies) that has sealed the game for them."
Parkinson made two changes for the Sheffield United fixture, with Frenchman Christopher Routis replacing Gary Liddle as City's holding midfielder and James Hanson making a welcome return from injury.
Parkinson said: "I thought Hans was excellent for us.
"He had only been training with us for a couple of days so we had to be wary of his match fitness going into the game.
"He's been doing a bit of running for a couple of weeks now so it wasn't a real gamble about his overall fitness but we knew that 70 minutes would be about enough for him."