After a goalless draw in the original tie down at the EBB Stadium, two second half goals from Parkinson's men in the replay eventually saw City over the line.
At a rainy and wind-swept Coral Windows Stadium, a well-taken strike from Greg Leigh just past the hour mark and a penalty from Tony McMahon fifteen minutes later proved enough to book City a date with Evo-Stik League Southern Premier Division side Chesham United.
The result extended City's recent unbeaten run to eight games in all competitions and also saw the Bantams keep a fifth straight clean sheet for the first time since 1954.
Parkinson said: "We are very pleased to get through into the next round. It was a difficult match, as we knew it would be, because Aldershot raised their game again as they did in the first tie.
"In the first period we were a bit hurried in our possession, almost as if we wanted it a bit too much, but the second half we were better.
"We played with more calmness and showed our quality at times.
"It was a great pass from Billy (Knott) for the first goal and Greg's first touch was tremendous - a really positive one. He's rounded things off nicely with a great finish as well.
"It was tough conditions out there in the wind and the driving rain - the second time in four days we've had to play in those sort of heavy conditions - so credit to the lads for keeping going.
"This is a cup competition we have some great recent memories in and we want to make some more."
The only concerns from the night for the Bantams was the sight of Gary Liddle and James Hanson both limping off the field in the second period.
Parkinson believes Liddle should be fine after taking a knock in the game but City will have to check on Hanson's twisted ankle.
Parkinson added: "Lids is ok we think - he's just got a whack on his calf.
"Hans twisted his ankle so we just hope that isn't too bad. We probably won't know the full extent, however, until we check again in the (Thursday) morning."
Following on from the cup replay win over Aldershot, City now face the prospect of a quick turnaround and a trip to Scunthorpe United on Saturday - the first of three league games for the Bantams in eight days.
Parkinson continued: "Saturday is another big game for us and we have to quickly recover from this one (vs Aldershot).
"That means getting the rest periods and the training load right for the players between now and the weekend.
"The trip to Scunthorpe will be a real test for us but we want to keep our recent run going."