The Premier League club announced the departure of De Boer today following a five-game spell.
In that time, the former Ajax boss oversaw one victory in the EFL Cup, but four defeats in the Premier League led to his downfall.
Attention has now turned to who will replace De Boer, with former England manager Roy Hodgson widely expected to take the role, but it has been confirmed that Allardyce was contacted by the club's chairman.
The 62-year-old spent seven months at the London outfit and helped retain the Eagles' top-flight status before stepping down at the end of last season to take a break from football.
As a guest on Sky Sports' Monday Night Football show ahead of the match between West Ham United and Huddersfield Town, Allardyce revealed that he has spoken to Parish.
"It's quite strange than I'm on this programme and Frank has unfortunately lost his job," said Allardyce. "Just to make it clear to everybody and to stop the speculation, I spoke to Steve late this afternoon just before I got here actually. I said for me I wouldn't be ready to come back.
"I made a decision and I wouldn't feel comfortable at this moment in time at taking any position as a football manager as I'm enjoying my life too much at the moment watching everybody else with the pressures of the Premier League.
"He didn't offer me the job but he just said 'what are your thoughts?'. Obviously I told him what I was thinking if you were to offer me the job, I think at this moment of time I wouldn't want to go back in."
Palace will face Southampton, Manchester City and Manchester United in their next three league fixtures.