Cole and Reid were taken off in the closing stages of Saturday's league win over Rochdale due to tight hamstrings.
The pair both certainly played their part in City's impressive win at Spotland Stadium but Parkinson says the duo will need to be right and ready to play if they are to feature in his plans for the second round tie.
Parkinson said: "Both with Devante (Cole) and (Kyel) Reidy, they just have tight hamstrings from Saturday.
"We will have to monitor them ahead of Tuesday night.
"There is nothing serious going on with either of them but we just have to make sure they will be ok if we wish to select them.
"With Kyel, we probably played him ten minutes longer than we would have liked but we left him on because he was playing so well."
Despite Parkinson being at the club now for over four years, this Tuesday's tie at the Coral Windows Stadium will be his first in charge of City in the Johnstone's Paint Trophy on home soil.
City's previous home match in this competition came in late August 2011 when Parkinson watched from the stands after officially being announced as the new Bantams boss the day before.
Parkinson added: "The first game here when I came into the club was the Sheffield Wednesday tie in front of the Sky Sports cameras.
"Colin Cooper took charge of the team while I watched on from the stands.
"Strangely enough, we've been drawn away ever since.
"Being at home this time around, we are clearly looking forward to it.
"We have some players coming back into the reckoning because of the lads who are on international duty.
"We will have to make some changes but the team we pick will still be one we feel can go and get us the result."