The tabloid reported on Tuesday that Parkinson's job is on the line, despite leading the Bantams to the League One play-offs this season, with the club's new German owners keen to appoint ex Leeds United boss Uwe Rosler.
Derby, who also failed in the play-offs, are on the lookout for a new boss with Darren Wassall's temporary spell due to end this summer.
Forest, meanwhile, have been managerless since March, after relieving Dougie Freedman of his duties.
A move to either club is likely to appeal to Parkinson, with Derby possessing one of the division's best squad on paper and Forest reportedly on the verge of fresh investment from Greece.
Blackburn, on the other hand, find themselves in a difficult situation with Paul Lambert tendering his resignation last month and several first team players due to depart on free transfers.
Relegated Bolton have also been without a manager since March and recently survived going out of business at the eleventh hour.