Bradford, playing in the fourth tier of English football, went through to the last four on Tuesday after beating the Premier League side 3-2 on penalties at a packed Valley Parade after the sides had finished 1-1 after extra time.
Arsenal had needed an 88th minute equaliser from Thomas Vermaelen to force extra time but despite putting Bradford under immense pressure they could not finish their numerous chances to kill the hosts off.
Bradford, who last played in the English top flight from 1999 to 2001, had won their last eight shoot-outs and made it nine in a row when Vermaelen missed Arsenal's final spot-kick.
"To be only three minutes away from beating a team of Arsenal's calibre says it all," Parkinson told the club's official website.
"The talk after the game might be about the penalty shoot-out win and our run of penalty victories, but that shouldn't take anything away from the lads' performance."
Without a trophy since winning the FA Cup in 2005, Arsenal put out a strong team that included Gervinho, Aaron Ramsey and Lukas Podolski but were caught cold when Garry Thompson put Bradford ahead with a side-foot finish in the 16th minute.
Arsenal took their time to get to grips with the buzzing hosts but almost made it 1-1 when Francis Coquelin hit the woodwork in the 38th minute before Gervinho squandered a glorious chance to equalise with the goal gaping.
Arsene Wenger's side were on top throughout the second half and the pressure paid off in the 88th minute when defender Vermaelen nodded home.
After a scoreless extra period, Santi Cazorla and Marouane Chamakh missed with the visitors' first two spot-kicks and Bradford's Ritchie Jones and Stephen Darby also failed to score before Vermaelen hit the post.
"This result means everything to us, that's the players, management and staff," added Parkinson. "We want to try and put Bradford back on the map for the right things, and nights like tonight will certainly help do that.
"The people of this city have been superb with the club, their support tonight was magnificent, and they deserve a team they can be proud of. I believe we are getting there with that one."
Also on Tuesday, Aston Villa booked a place in the last four by coming from behind to beat Norwich City 4-1 away.
In the other quarter-finals, Swansea City host Middlesbrough later on Wednesday while Leeds United will have to wait until December 19 to host Chelsea due to the Blues' Club World Cup commitments in Japan.
