Aaron Mclean's scrambled goal eight minutes from the end of normal time was enough to secure City's safe passage into the second round and make it two wins out of two following Saturday's opening day win over Coventry City.
With as many as seven teams losing out to lower league opponents during other first round ties this Tuesday evening, Parkinson felt the Bantams could be happy overall with their night's work.
Parkinson said: "We've had Morecambe watched over pre-season and down at Dagenham & Redbridge on Saturday and we knew tonight (Tuesday) was going to be tough.
"Morecambe are a side who I feel can be pushing at the top end of League 2 this season and they will have been looking to raise themselves tonight against a side at a higher level.
"I felt we handled the game well, however, in some tricky conditions with the wind and the rain.
"Regardless of whether tonight was a cup tie or not though, we wanted to show that this group of players has the mentality to go away from home and grind out the right result.
"We are learning about the team all the time but we seemed to have a resiliency about us. That is something we are going to need throughout the season.
"Playing a game away from home, you sometimes need different qualities to when you are playing at home, and I felt we showed those qualities in the game tonight.
"We didn't always show the same sort of composure in and around the final third like we did on Saturday (vs Coventry City). But when perhaps you don't have that as a side, you need your back four and goalkeeper to stay solid, and I thought they did that really well.
"I am pleased for Aaron as well to get the winner.
"The goal has perhaps looked a bit of a scramble in the end but he has showed great awareness and movement to enable him to latch onto Billy's (Knott) cross but also still stay onside.
"The win will keep confidence levels high. We still have lessons to learn as a team but we will look to keep developing as a team as the season progresses on."