Substitute Coventry striker Leon Best grabbed a dramatic late equaliser as former manager Iain Dowie made his first return to Selhurst Park.
Best darted to the near post to flick Jay Tabb's low left-wing cross beyond Julian Speroni on 87 minutes.
Palace took a first-half lead thanks to Stuart Green's classy strike after good work by striker Dougie Freedman.
Dowie was given a hostile reception before kick-off after his acrimonious departure to Charlton in May 2006.
Crystal Palace manager Peter Taylor:
"I am not confident (of keeping my job) but I just don't know because I have not spoken to (owner) Simon (Jordan).
"If I'm honest, it's not nice when you read that he has been having regular
contact with Neil Warnock.
"I hope I am given time because I enjoy the job. The position we are in is not good enough, but I don't think we are that far away from being a good side."
Coventry City manager Iain Dowie:
"Nobody can take away what I did here and I'll let other people spout off in the media.
"I never see Simon Jordan's name in the paper unless he is battering someone. I think when people look back in 10 years' time they'll see that I did okay.
"We got promotion in my first season, got relegated on goal difference, and then got into the play-offs again in the third year.
"They have made a lot of money on the players I left them. They also spent £8m on players last season and they didn't get to the play-offs."
Crystal Palace: Speroni, Butterfield, Hudson, Cort, Craig, Green, Watson, Fletcher, Soares, Scowcroft, Freedman.
Subs Not Used: Flinders, Lawrence, Idrizaj, Martin, Dickov.
Goals: Green 26.
Coventry: Konstantopoulos, Hall, De Zeeuw, Ward, Borrowdale, Cairo (Thornton 70), Stephen Hughes, Doyle, Tabb, Adebola (Best 62), McKenzie (Simpson 72).
Subs Not Used: Marshall, Hawkins.
Booked: De Zeeuw, Ward, Doyle, Cairo, Hall.
Goals: Best 87.
Ref: Andy Woolmer (Northamptonshire).