Parkinson snapped up his sixth signing of the summer by securing the services of former Huddersfield Town and Leyton Orient defender Nathan Clarke on a one-year deal.
Following his move to the Bantams, Clarke played the first half of City's pre-season friendly at Whitby Town.
Ahead of the game, City skipper Stephen Darby, who has been struggling with a stomach issue recently, managed to take part in the warm-up with the rest of the squad.
Parkinson admits that he was tempted to give Darby some game-time during the Whitby friendly but he decided to play it safe with the trip to Swindon Town for the season opener in mind.
He said: "Everything he (Darby) has done so far, he has come through unscathed. There has been no reaction around his groin area.
"That is a big relief for us.
"We were contemplating giving him 15 minutes tonight (vs Whitby) but we thought it best not to rush it.
"We've come so far with him and got him back into a good place in terms of the injury.
"We just want to keep building him up over the next ten days and see where we can get him fitness-wise."
Speaking about the arrival of Clarke, Parkinson added: "We clearly needed another centre back in the building, and Nathan is a very experienced player at this level. He has captained sides in League 1.
"He is a good talker, he's a leader and I am pleased to add him to the squad."