And Parkinson is set for new contract talks at Valley Parade later in the week.
The Bradford manager's current deal expires at the end of the season and there could be no better way of opening negotiations with Bradford's joint chairmen Julian Rhodes and Mark Lawn than on the back of another giant-killing upset.
The Bantams captured the hearts of the nation when holding their nerve to send Arsenal crashing out in an unforgettable penalty shootout in the quarter-finals and Parkinson, whose League Two side had already seen off Wigan, is ready to make the city proud again.
"I'd really like to stay at Bradford," said the 45-year-old former Colchester, Hull and Charlton manager.
"I had a brief chat with the chairman last week and we're going to have a sit down hopefully this week and try and get my, and the staffs', contract situation resolved.
"I think that's important because we want to bring players in and we've got a lot of players out of contract, so we're going to try and get that resolved sooner rather than later.
"I'm enjoying it here, it's a great club and we're starting to build something. Me and the staff feel that if we could get this club going it could really take off and we're just at the start of that."
Parkinson, installed as Peter Jackson's permanent successor in the summer of 2011 with the remit of steering the club out of League Two, rebuilt the whole squad during the close season and, but for a 2-0 defeat at Barnet on Saturday would be sitting in a play-off position.
Bradford are still only three points off an automatic promotion spot and Parkinson added: "Last year when we came in it was a real struggle to keep the club in the division and that was pressure. Talk about pressure (tonight), it's certainly a night to enjoy when you look at what we went through last season."
The Bantams flirted with relegation and finished 18th, six points clear of the drop zone, before Parkinson embarked on a complete overhaul.
"I was really pleased to get to the summer and change things round," he said. "Everybody's really positive in the city behind the club again and that's what we need.
"If you look at the crowds we get in our division, 10 and 11,000, that's quite extraordinary and the fans we take on our travels.
"We went to Wigan and took over 5,000. We've had a full house against Arsenal, so the potential is there to say this club is a very good Championship club.
"You've seen it in the past, and Bradford have done it themselves and got in the Premier League. It could certainly hold its own in the Championship, no problem."
It has been a long 13 years since Bradford escaped relegation on the final day of the 1999/2000 season to survive a second campaign in the Premier League after a famous 1-0 home win over Liverpool.
The Bantams were relegated the following season and then followed a calamitous slide into administration (twice) and English football's basement division. From top to bottom in seven forgettable seasons.
But not only were Arsenal conquered in the previous round, top-flight Wigan also succumbed to Parkinson's penalty shoot-out kings, while Gianfranco Zola's Watford were undone by two late goals at Vicarage Road in round two and Parkinson is ready to give the whole city something to cheer again.
"That's really important for us,"
he said. "It's a big club, a sleeping giant and we're trying to reawaken it.
"The win against Arsenal certainly reminded everybody that Bradford City is up and running and starting to get going. Another really good performance would be great for the city again."