A first minute strike from Jason Kennedy gave the Bantams the half-time lead but three unanswered Barnsley goals after the break saw the hosts pick up a deserved victory in front of the Sky Sports cameras.
Barnsley's second half comeback began less than four minutes after the restart when a wide free-kick delivery from Conor Hourihane found the bottom corner of Jordan Pickford's goal via the head of an unfortunate Rory McArdle.
Parkinson was clearly unhappy with the timing and nature of Barnsley's equalising goal but he felt the Bantams could have no real complaints about the game's overall final outcome.
Parkinson said: "We obviously got a great start but it was almost as if there was a feeling that we've had a good run away from home, they've had an indifferent start at home, it'll just take care of itself after the goal. I just got that feeling.
"I said to the lads at half time that I felt we had become a little self-indulgent in possession rather than playing as a team. We had some good patches of play in that first half, don't get me wrong, but Jordan (Pickford) had too many saves to make as well.
"Even though we had the lead at half time, we knew we had areas to improve but unfortunately in the second half we were second best.
"The home crowd were really quiet in the first half but when we gave away a soft goal at the beginning of the second period, it gave them a huge lift. Their players responded to that and we didn't have an answer today (Sunday).
"That is obviously disappointing for me to have to say but I felt we neglected too many of the basics this afternoon. There were certain areas where we were slack.
"We didn't respond well enough after their second half goals. We did have some balls flashed across their six yard box - maybe on another day someone could have gone sliding in on one of them - but it wasn't to be.
"With the likes of (Rory) McArdle and (Andrew) Davies at the back, it was unusual for us to concede the goal like we did early after the restart. I believe there is some talk that there was a push on Rory but it was still a soft goal from a bog-standard sort of delivery.
"We've had a great run away from home until today and we've played some good stuff but today wasn't good enough from us over the course of the whole game."