Parkinson had as many as five trialists, plus out-of-contract attacker Raffaele De Vita, in his squad for Saturday's goalless friendly draw with Blackburn Rovers at the Coral Windows Stadium.
Two of the trialists were goalkeeper Matt Urwin and French central defender Christopher Routis.
Given the number of trialists currently within City's ranks, Parkinson has revealed that the Bantams will look, if possible, to organise another friendly fixture for either this week or next.
He said: "We're not happy with where the numbers are at the moment. I know we need more players to add to the squad.
"We need another goalkeeper to add to the group, especially as Jordan (Pickford) can't play in the cups.
"It needs resolving because, no disrespect intended, we don't want to have to go into the cups and throw a youngster in goal if he isn't ready for it.
"With Matt Taylor going, there is space for another defender in the squad as well.
"We are also looking to strengthen in other areas, such as the wide positions. I know where we need to add.
"We've got a few trialists in at the club and some of them didn't get on against Blackburn so we will try to arrange a game for this week or next."