Wright made a major impression in his first appearances for the club after joining on loan from Bristol City, but picked up the injury late in the 1-0 win over Oxford United on February 15th.
Parkinson subsequently confirmed that he was set to miss eight to twelve weeks of action.
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Wright is nevertheless working with physiotherapist Peter Brand to ensure his recovery is affected as little as possible.
As it stands, Wright could theoretically return to play for the Black Cats later in the season.
The EFL have postponed all fixtures until April 31st at the earliest, and the FA have now raised the possibility of the season extending deep into June and beyond.
The EFL have insisted that they are committed to resuming and completing the current season ‘when it is safe to do so’.
Parkinson also said that the enforced break had given Charlie Wyke an opportunity to continue his recovery from a back problem.
Wyke missed the 2-2 draw with Gillingham and though he featured in the second half of the 2-0 defeat to Bristol Rovers, he has subsequently had another injection.
“Pete Brand is working with the injured players – Bailey Wright has been doing work individually and is progressing well,” Parkinson said.
“Charlie Wyke has been able to do work for his back problem, he had another injection in that.
“We just have to make sure that all the lads are doing the right work, and they are all good professionals who will keep themselves in the best shape possible.
“Following the new Government guidelines, they are still able to go out to exercise once per day so they can go for a run or a bike ride or whatever.
“We just have to make sure that when we are able to come back, we have done as much as we possibly can in the circumstances.”
Wright is on loan from Bristol City but is out of contract in the summer and expected to leave Ashton Gate.