Do Rupert, A&A, Misunderstand the Parity Principle?

Menary writes,

These critics think that the main argument for the extended mind is simply the claim that if external processes are sufficiently similar to internal ones, then they are cognitive. Is this really the argument for EM? I believe that the critics have reached this conclusion by misinterpreting the PP.  (Menary, 2010, p. 6).

Clark has previously suggested that Rupert, Adams and Aizawa have been misinterpreting the Parity Principle, but I'm not really sure why he and Menary think this.  What have we written that leads them to think that the PP is the basis for our thinking that one argument for the extended mind is that external processes are sometimes sufficiently similar to internal ones?  Here is why I think that there is such an argument.  It comes from the Inga-Otto example (but also the three modes of Tetris play case.)  Note the following:

For in relevant respects the cases are entirely analogous: the notebook plays for Otto the same role that memory plays for Inga.  The information in the notebook functions just like the information constituting an ordinary non-occurrent belief; it just happens that this information lies beyond the skin (Clark & Chalmers, 1998, p. 13).

Certainly, insofar as beliefs and desires are characterized by their explanatory roles, Otto’s and Inga’s cases seem to be on a par: the essential causal dynamics of the two cases mirror each other precisely (ibid.)

Clark, A., & Chalmers, D. (1998). The extended mind.  Analysis, 58, 7-1

Menary, R. (2010). Introduction. The Extended Mind. Cambridge, MA: The MIT Press. (pp. 1-25)