First half goals from Billy Clarke and Jon Stead were added to by quick-fire efforts from the recalled Filipe Morais and Mark Yeates within less than a minute of each other just before the hour mark.
Dartford netted a consolation goal through midfielder Lee Noble shortly afterwards but Parkinson's men always seemingly had plenty in hand as they cruised over the finish line during the time that remained in their second round contest to register a comfortable 4-1 win.
City's victory over Dartford this Sunday meant the Bantams booked their place in the third round for only the second time since they were relegated from the Championship eleven seasons ago.
The only other time City had qualified for the third round during that period was in Parkinson's first season in charge at the Coral Windows Stadium during the 2011/2012 campaign.
On the same day when Chester and Worcester City earned replays away at League 1 clubs, Parkinson was delighted to see City show the right attitude when approaching their own encounter with non-league opposition, ensuring their afternoon against wannabe giant-killers went much more smoothly.
Parkinson said: "We approached this game in exactly the right way.
"If you look around the country today (Sunday), some of the non-league teams have done well and the games have been tight. This one wasn't, however, because of the way we approached it.
"We were spot-on in the way we went about the job. The only complaint we can have really is that we got a bit sloppy at 4-0 up and we conceded a goal.
"All in all though, it was a terrific performance from us and we looked a great threat going forward.
"No disrespect to Dartford, but we knew we had better players in our side then them and we knew that if we played properly and moved the ball well, we'd create chances, and that's what happened.
"These games can be difficult. People were coming to this game today expecting it to be easy for us but it could have been an anxious afternoon if we hadn't approached the game in the proper manner."
After seeing off Dartford this Sunday, City will now be joined by all of the country's Premier League and Championship sides in Monday's third round draw.
Parkinson added: "It is good to be in the hat - it really is.
"This is only the second time in eleven years that we've been in the third round draw - a club like Bradford City needs to be in this draw really.
"It is one of the most anticipated draws in the football calendar and, on Monday evening, we'll be in it. It'll be interesting now to see who will pull out of the hat.
"The F.A Cup is special, we wanted to be in the draw this Monday, we really did.
"A lot of the talk before this game was about how big this match was for Dartford but it was an important game for us as well. It is great for us to be in the draw now and the lads are delighted to be in there.
"On Monday night, we can sit back and enjoy the draw."
Following City's fourth goal of the afternoon through Yeates, skipper Stephen Darby was replaced by Christopher Routis at right back.
Parkinson confirmed after the tie that the substitution was made as a precaution to rest up Darby due to a minor calf strain.
Parkinson continued: "The two quick-fire goals in the second half really killed the game off so we brought Stephen off because his calf was a bit tight.
"He didn't want to come off but we felt it was the right thing to do so we could protect him for next weekend's game at Chesterfield."