I agree with you that the definition of "parsing" is not well-defined in the context of this thread. As I have mentioned elsewhere in the thread, in Perl the line between "parsing" and "execution" is fuzzy and in my opinion artificial. The distinction is influenced from the way other, more conventional programming languages are executed, which doesn't apply well to Perl.
The OP has convincingly proving that it is undecidable to determine the prototype of a given sub in a given piece of code. The choice of problem is because (now this is an appeal to intuition, not a formal statement) however you define "parsing", determining the prototype of a sub is something that by all sensibilities should result from "parsing". (Update: the prototype of a sub can also be made to affect whether a "/" is interpreted by the compiler as a division operator or the start of a regex match). Elsewhere I listed the example of determining whether a module loads successfully or not (i.e., whether the module returns a true value) as another thing you might imagine as being an outcome of "parsing".
So I can see where the OP is coming from, but of course it is a bit moot if you reject the artificial distinction between parsing and execution in a language like Perl, which allows arbitrary execution to be freely mixed with "parsing" phases. When you rephrase it more realistically as (for instance) "the BEGIN phase of Perl is Turing-complete", the whole exercise is trivial and loses much of its shock value.