Parkinson, 45, takes his side back to Wembley this weekend in the npower League Two play-off final against Northampton, three months after guiding them to their historic Capital One Cup final appearance.
Victory for the West Yorkshire side on Saturday would cap one of the most memorable seasons in the club's history, on a par with promotion to the Premier League 14 years ago, and represent the first step on the ladder back to the top flight.
Swansea gave Bradford a footballing lesson in the Capital One Cup final in February, but did well to remember a similarly significant date 10 years earlier when they beat Hull on the final day in the old Third Division to preserve their Football League status.
The two clubs will meet again in the Premier League next season after Hull's thrilling last-day promotion, and when asked if Bradford could rise from the depths and emulate both, Parkinson said: "Without a shadow of doubt. No question about that.
"If we did get up I'm sure the season-ticket sales would be on a par with anybody else in League One, maybe not Wolves, but everybody else."
Bradford's players vowed to return to Wembley after Swansea had given them the runaround and lost only two of their remaining 15 league games to secure the final play-off spot.
"This club is a sleeping giant, there's no doubt about it," Parkinson added.
"It's a big ship to start turning round and I believe we're doing that.
"If it goes our way on Saturday it would be a massive step towards our progression, but a lot of progression has been made despite that.
"That's the great thing about football in this country. Teams can fall on bad times but you get a bit of momentum going, especially with clubs with a big fanbase, and anything can be achieved."
Parkinson is expected to commit himself to the club for another two years next week, but has delayed signing his new deal until assistant Steve Parkin and fitness coach Nick Allamby have agreed their new contracts.
Northampton manager Aidy Boothroyd believes his previous play-off final experience with Watford has stood him in good stead this time round.
Boothroyd led the Hornets to promotion to the Premier League with a resounding 3-0 victory over Leeds in the Championship play-off final at Cardiff's Millennium Stadium in 2006.
Having been there before, the 42-year-old thinks it has helped him to bring a relaxed frame of mind to the Cobblers camp.
"It is high pressure and it is nerve-wracking when you do it for the first time, but I'm a lot more relaxed about this one because I see the potential pitfalls we can get into," Boothroyd said.
"There are a lot of distractions in the build up to the game but the trick is to try to make it as normal as possible.
"Of course, it's not. It is high pressure. But you try to take as much away from the players and staff as possible.
"There are two things that are going to happen on Saturday. There is going to be an occasion and there is going to be a game. We have got to make sure that we play the game."