But, I think that part of what people think might be going on with the Parity Principle, in addition to the idea that in-the-head versus outside-the-head makes no cognitive difference per se, is something to do with recognizing processes as cognitive.
But, Clark and I disagree on what we should recognize as cognitive processes. Clark thinks that Martians manipulating bitmap images should count, but A&A don't. So, it looks like what might be worth investigating in the Parity Principle is what should count as a cognitive process. But, that seems just to be an indirect route to what A&A have loosely described as the mark of the cognitive, about which we have said a lot.
But, Clark has also expressed some scepticism about there being such a thing as the mark of the cognitive. So, I don't see what payoff there is in more attention to the Parity Principle. If it's about the MotC, then talk about the MotC.