The League Two club were denied victory in normal time when Thomas Vermaelen cancelled out Garry Thompson's early goal on 88 minutes.
And Parkinson hopes Bradford's progress to the semi-finals for the first time in their history will act as a catalyst for the club returning to the big time again.
"The lads were terrific, the commitment was first class and considering we were three minutes away.... People will talk about penalties but our overall performance was outstanding," Parkinson said on Sky Sports 1.
"Arsenal have got world class players and played their strongest team, but you have to give the players so much credit.
"I hope this can be the first of many nights like this for Bradford City.
"We want to put Bradford on the map again. I want them to have a team to be proud of."