How close can you get to the edge of the Parmesan before you start getting into the part of the rind you shouldn't eat?

A fine question.
The rind that your talking about is actually just hardened cheese. It gets this way from being exposed during the production process both to salt water: where it sits for 20-25 days and then to air: while aging for at least 12 months.(These are standards the real deal Parmigiano-Reggiano cheese has to meet, there are a lot of impostors out there. I spotted one just yesterday that boasted of being aged 90 days)

This rind is actually edible all the way out to the fine leathery skin. I like to save my Parmesan rinds in the freezer to throw in soups and stocks, they impart a great flavor.

A side note: it was once a rule that the whey left over from producing Parmigiano-Reggiano went to feed the pigs raised for prosciutto di parma.