The move, which will render Stead ineligible for Huddersfield for the remainder of this season's competition, was sanctioned by Terriers boss Chris Powell following a discussion with Parkinson earlier this week.
Middlesbrough loanee Andy Halliday has also been given permission by his parent club to play in the televised first round tie.
Jordan Pickford won't be allowed by Sunderland to feature in the West Yorkshire derby, however, meaning Ben Williams will start in goal.
Part of the back four to line up in front of Williams come Sunday could include Northern Ireland international Rory McArdle following the defender's return from suspension.
Parkinson said: "I am really pleased Jon (Stead) can play for us on Sunday because he's been doing well recently and we want him to carry it on."