Former Liverpool right-back Stephen Darby was Bradford's matchwinner with a goal five minutes from the end of extra-time to crown a dramatic comeback at a wet Valley Parade by the Bantams, who reached the fourth round in the competition for the first time in 24 years.
Parkinson said: "It's great to be in the hat for the first time for such a long time. Let's hope we get a big draw - everyone in the club deserves it.
"We played with commitment and drive and produced a terrific performance. Now the lads can celebrate."
Parkinson added: "We thought we were 2-0 down against the run of play after dominating the game and we said at half-time that if we can play in the same manner we will certainly get chances.
"I thought with the chances we created we deserved to finish the game in normal, never mind extra-time."