The Mirror says Parish has sounded out ex-Stoke City boss Pulis and is desperate to persuade the Welshman to take charge after preliminary talks went well.
Only if Pulis turns the Londoners down will they look elsewhere, with Neil Warnock - Palace boss from 2007-2010 - next in line to replace Ian Holloway, who stepped down by mutual consent last week.
Pulis is understood to have major reservations about the Selhurst Park gig.
He wants a bumper £1.5million-plus contract, plus a big survival bonus and fears it would be the hardest job of his career to save a side that's lost eight of the first nine games from relegation.
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