The Bantams triumphed 3-2 on penalties in the Capital One Cup to reach a first ever semi-final after the match finished 1-1 after extra-time.
The spot-kick specialists won their ninth successive penalty shoot-out 3-2 after Thomas Vermaelen struck the post.
The Arsenal skipper had earlier appeared to rescue the Gunners with a header three minutes from normal time to equalise Garry Thompson's first-half volley.
Parkinson said: "I don't think the enormity of our achievement has quite sunk in. When get up in the morning and see the headlines in the newspapers they will realise what they have done.
"There is a big gulf between the two teams in terms of finances and places in the divisions.
"I don't want the talk to be about the penalties, I want it to be about the performance. Arsenal have played their best team and we were three minutes away from beating them, then we have had the setback of conceding a goal and had to dig deep again in extra time.
"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."
- Sports & Recreation