What you say is true -- in fact I am writing a parser which does ambiguous parsing, and it was that which motivated the investigation which resulted in this proof.
in reply to Re^3: Perl Cannot Be Parsed: A Formal Proof
in thread Perl Cannot Be Parsed: A Formal Proof
However the claim is that Perl parsing is undecidable -- and listing the possible decisions does not make it decidable. Coming up with all the multiple parse trees may produce an answer adequate for most, or in particular cases, all practical purposes. But it does not decide the parse.
Similarly, we deal with the halting problem adequately for practical purposes with every program we write. However, the halting problems remains undecidable, and that remains a useful thing to know as we in practice write 1000's upon 1000's of programs which must and do halt.