I think Parker has very little ability as a manager, similarly to Lampard. The chatter seems to be that it wasn't the result as much as Parker's comments after the game that got him sacked. He may have avoided the boot had he kept his thought's to himself about them being ill-equipped for the level. Not wrong though was he.
Rodgers is teetering, but the growing talk out of Leicester is that the pandemic has smashed the finances of the ownership and they would sell for a reasonable offer. Rodgers also has decent time left on his deal and is highly paid - Leicester's ownership are clearly unwilling to invest much at present and will probably persist with Rodgers unless it looks likely to cost them their place in the Premier League, which would hugely devalue the club.
Conte as ever is a threat, but as was pointed out above, to walk, not to be sacked. He's the best manager they've had in a long time, but he's only this week saying Spurs need to invest more, haven't they spent over £100m this summer, net? This is classic Conte. At Inter, literally days after they shattered their transfer record to drop £70m on Lukaku he was claiming he can't work with the squad unless they spend more. Levy won't put up with this long term.