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.