*This is really a comment, but it's too long:*

I'm not entirely sure what you're asking, but I think there are a couple things worth pointing out:

 - In terms of what a model $M$ "knows" about a structure Ackermann-coded into one of its subsets $A$, note that this question best makes sense if $M$ knows about $A$ to begin with. But most of the time, the $A$s we'll be interested in won't be definable in $M$: indeed, no model $W$ of ZF can be Ackermann-coded into *any* nonstandard model $M$ of PA in a definable way (since such an encoding would give evidence *in $M$* of the non-standardness of whatever element of $M$ is coding the $\omega$ of $W$). So you have to be very careful when you ask "What does $M$ know?" about such a situation - *what exactly do you mean*, given that the most basic information about such a configuration will generally be undefinable in $M$?

 - In terms of Parikh's model, I think you're overestimating its relevance to Ackermann coding. Any model of PA has a unique definable map satisfying the recursive properties of exponentiation. *(Indeed, PA proves "there is a unique exponentiation function" in the following sense: PA proves "for all $n$, there is a unique map from $\{0, 1, . . . , n\}^2$ to $\mathbb{N}$ which satisfies [recursive properties]". Note that such a map is a finite object, so can be talked about directly in PA, even though the whole exponentiation map can't.)* Ackermann coding works with respect to this definable map. Parikh showed that we can have *undefinable* maps that also satisfy the properties of exponentiation. But these maps aren't relevant to Ackermann coding.