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The Lancashire Telegraph said Parkinson, 42, had suffered a stroke.
A Blackpool statement read: "Head of youth Gary Parkinson is in an extremely critical condition."
The statement on the official club website continued: "The thoughts of all at Blackpool FC are with Gary's family at this extremely difficult time."
Parkinson began his career as a trainee at Everton, before moving to Middlesbrough in 1986, where he played more than 200 games.
After brief spells at Southend and Bolton he switched to Burnley, where he is best remembered for scoring the winning goal in the 1994 Division Two play-off final against Stockport.
Parkinson's last significant career move was to Preston, where his appearances were restricted by injury. He joined Blackpool in 2001, playing 24 games, before returning to Bloomfield Road as the head youth coach in 2006.
A Burnley spokesman said: "The club is very sorry to hear this news. We send both Gary and his family our best wishes at this difficult time and hope he can battle through."
A spokesman for Preston added: "Our thoughts go out to Gary and his family at this extremely difficult time."
Source: BBC Sport
Source: BBC Sport