A circumstantial clause describes the state of its ṣāḥib al-ḥāl and has an accusative syntactic position, whether it carries a linking wāw or pronoun. A parenthetical clause, by contrast, does not occupy such a grammatical slot in the host construction.
ExpressionPrimary criterion
Circumstantial clauseDescribes a state; accusative clause-position
ParenthesisAdds an aside; no position in the host syntax
The presence of wāw alone cannot decide the category: syntax and rhetorical purpose must both be examined.