The Bantams came from 2-0 down to stun the Barclays Premier League leaders 4-2, with a Bradford side which cost Â£7,500 in transfer fees beating Jose Mourinho's Â£200million team of superstars.
It was just the latest in a series of Bradford cup upsets in recent years, and the manager now fancies another big name in Monday's fifth-round draw.
"If I had my choice, I'd like to play Manchester United," he told Sky Sports News.
"The experience we had yesterday going to Stamford Bridge was just terrific
If Manchester United can manage to beat Cambridge it would be great to go to Old Trafford
"But whoever we get, it's great to be in the fifth round
We're enjoying being in the limelight."
Bradford's win headlined a day of stunning upsets, with Manchester City, Tottenham and Southampton all also out a day after League Two Cambridge held United to a 0-0 draw on Friday night.
Parkinson is an expert in cup upsets after his side beat both Arsenal and Aston Villa to reach the Capital One Cup final in 2013, and said the Chelsea win was his best yet, even if Mourinho made several changes to his side.
"The night we beat Arsenal they played a full-strength team but it was a night set up for a cup shock with a pitch starting to get a little bit frosty," he said
"It was a freezing cold night in Bradford with all the elements for a cup shock.
"Going down there to Stamford Bridge, with a fantastic playing surface and 40,000 people behind them, to beat the league leaders and one of the best teams in Europe has to top the Arsenal victory."
Chelsea took a 2-0 lead through Gary Cahill and Ramires, but Jon Stead pulled a goal back before half-time before goals from Filipe Morais, Andy Halliday and Mark Yeates completed the upset
"It's really just about giving the lads the belief to go out there and put on their best performance," Parkinson added
"We respected Chelsea but we certainly didn't fear them
We weren't going to back off them and give them the ball in our half
When they came into the middle third we pressed them aggressively and that was the key
"We didn't give them any time on the ball and they're not used to that at that level
When we had the ball ourselves we played with real calmness and I'm really pleased because it's so important
These days don't come along very often and when they do it's great we made the most of it and the lads really did themselves credit."
Mourinho described Chelsea's performance as a "disgrace" but came into the Bradford dressing room to congratulate Parkinson and his players after the match, a move which Bantams goalkeeper Ben Williams said was the "icing on the cake"
"It was fantastic
The scenes, the look on everyone's faces was amazing, something which will live long in the memory," Williams told BBC Radio Five Live's Sportsweek programme
"The fans were amazing and after that Mourinho came into the dressing room and shook everyone's hand and congratulated us, that was a surreal moment as well.
"In he walked and the celebrations stopped
This figure waltzed into the dressing room
He was very classy, very humble, very respectful - it was the icing on the cake."
It was also an extra-special day for Morais, who began his career under Mourinho at Chelsea but never made a first-team appearance, leading something of a nomadic existence since
The 29-year-old struck the equaliser 15 minutes from time and told the London Evening Standard: " It's things we dream about as players
"I've dreamt for a long time about coming here and finally I got my chance
I dreamt of scoring a goal at Stamford Bridge as a kid
I've come back as a Bradford player and done it and it's a great achievement for me and the team
"The chance falling to me like that was purely a gift from God because that just doesn't happen."
Source : PA