Goals from United trio Billy Sharp, Conor Sammon and David Edgar were enough to seal the points for the hosts at Bramall Lane, while Gary Liddle netted a late consolation for the Bantams seven minutes from time.
Despite Sharp giving the Blades the lead at the interval thanks to an eleventh-minute opener, Parkinson's men were still very much in the contest at the half time break following an encouraging first period display.
Two goals in the space of four minutes towards the start of the second period from Sheff Utd, however, left City with it all to do.
Parkinson said: "There was nothing in the game in the first half and I fancied us quite strongly to go on and do something in the second period.
"We did say to the players that Sheff Utd do have a lot of experienced players though, and we would have to be wary of them starting the half quickly attacking towards their main (Kop) stand behind the goal.
"We had to be ready for the pressure they were going to put us under. In the period up until their third goal though, I felt we creaked a little bit as a team, and that is the first time I've had to say that about us for a long time.
"We went away from the basics of our game - getting tight to people, stopping crosses, defending balls in our box - and we were punished for it.
"In recent weeks, teams have had periods against us but we have managed to stand strong against them and go on to pick up some good results.
"If we could have got through that period today (Monday), we would have had chances to get back into the game later in the half.
"We have to learn from this game. We have to accept that teams will have periods of play and we have to stand strong against that like we have been doing in recent weeks.
"We had options to change it from the bench if we needed to try and get us back in the game, but that period after the break was very costly.
"That is hugely frustrating for ourselves and I'm sure for the supporters as well.
"They'll be going home disappointed after today's result, and we share that disappointment with them.
"We've had a good run of results before this game and we have to now dust ourselves off and come out fighting in our next match at Gillingham."
City's preparations for their trip to Bramall Lane were far from ideal, with the same rain showers that caused the postponement of their Boxing Day encounter with Burton Albion also putting their Apperley Bridge training ground temporarily out of action due to flooding.
Parkinson added: "We don't want to make excuses but there were times when we did some uncharacteristic things and we looked like a side who perhaps hasn't played too much football recently.
"Wherever we have to go in the next few days to find somewhere to train, we will make sure that we'll have had plenty of training time with the boys by the time we go to Gillingham.
"We know we have the ability to go on another run of results but we will have to give the players some reminders between now and Gillingham of what is required.
"Don't get me wrong, this (Bramall Lane) is a tough place to come - Sheffield United have a lot of talented players. They have the squad and the set-up worthy of getting them into the top two of this division.
"Wherever we play our football though, we have to do the basics right to give ourselves the best chance of getting some points. For that period in the second half, we didn't do that."