*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. Ackermann coding relies only on being able to prove that binary expansions of natural numbers exist and are unique; this is ultimately only a statement about the map $x\mapsto 2^x$. This is legal, since - in any model $M$ of PA - we have a *definable* exponentiation map, which PA proves satisfies the recursive properties of exponentiation. The possible existence of other maps which satisfy the recursive properties of exponentiation isn't relevant, *since those maps won't be definable*: any two definable maps satisfying the recursive properties of exponentiation must be the same, by induction.