After defeating Chelsea 4-2 at Stamford Bridge in the previous round, Bradford made the most of home advantage on Sunday as they progressed to the last eight for the first time since 1976.
John O'Shea's early own goal and Jon Stead's effort shortly after the hour mark earned them victory at Valley Parade, and an ecstatic Parkinson hopes they are given another home tie when the draw is made on Monday.
"I am obviously delighted with the way we played," he told BBC Sport. "We did the right things in the right areas of the pitch.
"Defensively, we were outstanding today. I was pleased when Jon Stead put that second goal in, it gave us a bit of breathing space and we saw the game out professionally.
"There is always a danger because the expectancy might have been we would win this one after beating Chelsea, but we have kept that underdog spirit. It was a good, old fashioned cup tie - a really good advert for the FA Cup.
"The pitch was not as bad as it was two or three weeks ago, we played some good stuff at times, so did Sunderland.
"After today we would like another game at home. It is incredible for all the players to experience that sell-out atmosphere.
"We had all three sides [of the ground] singing and behind the team and it raises the lads to the next level."
