Parish dismissed former Scotland boss George Burley after the New Year's Day defeat at Millwall, with former Palace striker Dougie Freedman put in temporary control of the Selhurst Park club.
Coppell, who has had four spells in charge of the Eagles, has been out of work since quitting as Bristol City manager just two games and three months into the role in August.
The 55-year-old, who has also managed Manchester City, Brighton and Reading, watched Monday's 1-0 win over Preston and Parish is keen to explore the possibility of bringing him back for a fifth time.
"I don't know what Steve's situation is, whether he wants to manage or whatever," Parish said after Monday's game. "I absolutely will be giving him a ring and having a chat with him, Steve would be a great sounding board for us in terms of who we should get, he cares for the club deeply.
"The word legend is used too often but in his case he's a definite legend here. He's been a fantastic manager at almost every club he's been to and he's definitely someone who I would be mad not to have a chat to."