I think it depends on what Sage is saying to Parish in private.
At one extreme, if Sage is saying 'Steve, I play pressing football because God told me to and I'd rather drink paint than change. That's what we are doing. If these idiots can't get it right then that's their fault' then I imagine Parish will be nudging Hobbs and whispering 'See if Dychey still wants the job...'. If the word 'Philosophy' has left Sage's mouth then all bets are off.
At the other extreme, if Sage says 'Steve, I'm trying to get on the front foot and have a go, but if it keeps coming unstuck like this then, after a few more goes at it, I'll have to change to a far more risk-averse method, if only for now' then I'd guess Sage is as safe as safe can be, as that change in approach is all Parish would achieve by replacing him with Dyche anyway.
More realistically, if Sage is saying 'Listen Steve, you hired me knowing I play pressing football. That's how I did so well in France. I believe it can be effective but it takes time to implement, and I haven't had that time given how late the squad was actually together and at my disposal. Give it time, it'll work' then Parish will have serious food for thought.
Basically, I think a lot will turn on whether Parish is convinced by how Sage explains his thinking.
You'd assume that would be a big part of the job interview process. Outlining his philosophy but setting expectations on how quickly and perfectly it can be implemented.