John Windass scored the winner as the Reds edged a tight contest with Championship promotion rivals Ipswich 1-0 at Stok Cae Ras.
It was another notable victory for Wrexham, who are back in the second tier after securing a record-breaking three successive promotions, as their rise under the ownership of Hollywood stars Ryan Reynolds and Rob Mac continues.
Reds’ boss Parkinson said: “We spoke before the game about our chance to create a bit of history.
“We’ve done it and I’m so pleased because it was always going to be a tough game against a very good side.
“Every season in football is special and you’ve got to savour it, and the FA Cup is part of that.
“It was important that we backed up the Nottingham Forest win with another win.
“This season, with the challenge of being in the Championship and now an FA Cup run, we want to enjoy it, and tonight we certainly did.”
This week marked the fifth anniversary of Wrexham’s takeover, which took place when the club were in the National League.
They have been promoted in the past three seasons and, currently sixth in the Championship, they now have a shot at reaching the Premier League.
Parkinson said: “I think it’s great for our owners, their first year anniversary, to get them into the fifth round.
“We can really look forward to the draw on Monday night when we’ll be in the hotel in Bristol.
“We’ll savour that but, obviously, it’s the back-burner now for the FA Cup and we’ll concentrate on the league.”
Ipswich manager Kieran McKenna made 10 changes, with the Tractor Boys currently third in the Championship and an immediate return to the Premier League clearly his priority.
Wrexham are in contention for a play-off spot and although continuing the cup run means more fixtures to play and the schedule even busier, Parkinson doesn’t see it as an issue.
“There’s a lot of talk in football of there’s another game and now we’re through,” he added.
“Great, that’s what we’re in football for and we’ve got to embrace it.
“We’ve got a squad of players, we’ve got players who didn’t make the bench tonight and by the next round of this competition, we should have another three players back into the fold for that.
“I want to savour it and every game I want to enjoy this season. Every season but this season, the challenge of being in the Championship and now an FA Cup run, we want to enjoy it and tonight we certainly did.”