Which of the following sentences uses the correct pronoun forms and why?
My parents met in Havana -- him an artist and her a singer.
My parents met in Havana -- he an artist and she a singer.
The adjunct is a verbless clause. The choice of case for the subject of such an adjunct is a matter of style: in your example, the nominative "he"/"she" being the formal variant, accusative "him"/"her" the informal.
The clause, although verbless, nevertheless contains a predicative element (cf. "He is/him being an artist and "She is/her being a singer".)