Second half goals from James Hanson, Billy Knott and Mason Bennett were enough to send Parkinson's men on their way to a comfortable win, extending City's unbeaten start to the new campaign in the process.
Parkinson said: "It was a good all-round performance and three terrific goals from us.
"We controlled the game for long periods and even responded well when they scored their equaliser.
"We could have scored more to make the score-line even more convincing - we've hit the bar through JK (Jason Kennedy) - but overall I felt we were excellent. There was a real authority and intensity in our play.
"We spoke to the lads in the hotel last night about how we have played in the last season or so when we have got wins away from home.
"Virtually every one of them has seen us play like the home side - play with really commitment and desire - and I felt we showed that again tonight (Tuesday).
"The lads were champing at the bit before kick off tonight - you could sense there was a real determination to get the job done - and they have got their rewards."
Hanson's opener was his third of the season already for the forward. The goal was also the first Crawley had conceded all season.
Knott and Bennett's goals were their first for the club.
Parkinson added: "If you look at all the goals individually, they are all good goals in their own way.
"Darbs (Stephen Darby) and Jason Kennedy have shown great effort and drive to link up together down that right hand side. Once the ball is in the box, it is a great finish from James. I don't think some people are quite aware of what a good finisher he is with his feet.
"The way he has got the ball out of his feet, controlled it, and then got his shot away was very, very good.
"For the second one, it was a quality finish from Billy Knott after some good link-up play between him and Filipe Morais. Once Mason was in for the third, I think we were all confident he would stick it away - he looked very calm in that situation.
"He has a great calmness about him when he is in front of goal."
Bennett was only on the pitch after coming on at half time to replace Aaron Mclean, who was handed his first start of the season at the Checkatrade.com Stadium.
Rather than the substitution being about a potential injury though, Parkinson said it was more to do with Mclean's general match fitness after the forward missed most of pre-season with a calf problem.
Parkinson continued: "It was a decision we made pre-match. We have decided that we can only give Aaron his pre-season in the first team.
"We felt 45 minutes for him was enough tonight.
"He looked fresh and he was playing really well in the first half, and I was contemplating keeping him on at half time, but I just felt we didn't want to ruin his work for the night by keeping him on 15 minutes longer and risk him picking up another injury."
With Peterborough United due at the Coral Windows Stadium this Saturday, Tuesday's Crawley victory ends a run of three straight away games for Parkinson's men.
The Crawley win for the Bantams, coupled with their Capital One Cup win at Morecambe and goalless draw at Walsall, means City came away from those fixtures with two wins and a draw and only one goal conceded.
Parkinson said: "We have made good progress and learned a lot from these three games. We must not get too carried away though and we must still look at this (Crawley) game, analyse it and see where we can keep improving.
"The preparation for Saturday begins now. We look forward to walking out at Valley Parade having picked up these good results on the road but we want another one this Saturday against Peterborough.
"You can never guarantee a victory but what we must look for is another performance like tonight. If we can produce that again then we'll give ourselves a good chance of getting something out of the game."