League One's Bradford trumped Sunderland 2-0 at Valley Parade to add the Wearside club to their list of scalps that also includes Chelsea and Championship club Millwall.
In the 2012-13 Capital One Cup, Bradford knocked off Premier League clubs Aston Villa, Arsenal and Wigan on their way to the final where they lost to Swansea City, and Parkinson, who was also in charge then, believes his team can reach Wembley Stadium again.
"If we get our approach right we have a chance against anyone," he said.
"We would love Man United. The chairman will want to be away at a big Premier League club, but I want to be at home.
"Here [at Valley Parade], we can beat anyone if we keep the honesty and do the right things, play as a team. It is about individuals looking good in a team unit with no egos and then getting the rewards.
"The excitement will intensify now in the next round because we are one game from Wembley. We will give it everything, try to attack at every chance and walk off with our heads held high."
United must overcome another League One club - Preston North End - on Monday if Parkinson is to get his wish.
Bradford's "approach" was hardly revolutionary but was too much for Sunderland.
"We started really quick and we were physical in terms of closing down and challenging," Parkinson said.
"We set out to make it uncomfortable. We played with that underdog mentality and maybe we did catch them by surprise."
