The notion of a stochastic parrot is just a subset of this. It is what I'm arguing against, not for, on the basis that talking about "just" a statistical model

The notion of a stochastic parrot is just a subset of this. It is what I'm arguing against, not for, on the basis that talking about "just" a statistical model is meaningless exactly because something being a statistical model does not imply it is just a stochastical parrot. As such it's a wildly misleading claim to make. Yes, we're more than parrots, but that does not mean we can be described equally validly as "statistical models". Suggesting something is "just" a statistical model is a statement that is close to semantically void. It tells us near nothing about the computational limits - up or down - of something. Even a very simple system for running a statistical model with a loop around it providing IO can be made Turing complete.