In front of nearly 2,000 City followers at the weekend, Parkinson's men battled their way to a creditable goalless draw at the home of their South Yorkshire neighbours.
Attacking a vocal away end at Oakwell in the second period, City certainly finished the match on top and looked the more likely winners in the last 20-25 minutes.
Parkinson said: "I felt it was the best the supporters had been all season but obviously there is a reason for that. They saw a committed display from the lads and they responded to that.
"We've had 45 minutes spells at times earlier this season but this time we gave them proper performance throughout the whole 90 minutes.
"They were right behind us all the way on Saturday, and we've got to earn that support again - it doesn't just happen.
"They stayed with the lads all afternoon - it was like a home game in that second half attacking to that away end.
"They were tremendous, and I thank them for that support.
"They were like that, however, because they responded to a far better performance from us."