Parkinson's men are set to travel to Deepdale on their return to League 1 football this weekend looking to build on the back off last Sunday's F.A Cup win at FC Halifax Town.
Gary Liddle and James Hanson both missed the Halifax tie through injury, while Jon Stead suffered a slight groin strain during the match.
Speaking to Bantams Player in the build up to the Preston match, however, Parkinson had better news about the trio.
Parkinson confirmed that the three players had all been involved in at least some part of training today (Thursday) and, as a result, he refused to rule any of them out of making the Preston match.
The City chief said: "Things are looking better.
"Hans (James Hanson) has built himself up during the week and he has trained fully today (Thursday). That is clearly a big bonus for us.
"Jon Stead had a tight groin over the weekend but he has done a part of training this morning. Lids (Gary Liddle) has also been out there today.
"It is looking better for us, which is obviously a positive."
Following his recent call up, Oliver McBurnie will hope to pick up his first Scottish Under 19s cap this Friday during their friendly with the Netherlands.
Whether the young striker will join up with the rest of his City team-mates afterwards at Preston could depend on how much game time he gets with Scotland and the fitness of fellow forwards Hanson and Stead.
Parkinson added: "We've basically said that we will see how Steady (Jon Stead) and Hanson are tomorrow (Friday). Tomorrow will be a test for them to see if they will be fine for the weekend.
"If Oli (Oliver McBurnie) isn't involved much or he doesn't get any game-time at all for Scotland, we could still look to get him down here for Saturday to have him on the bench."