Parshin proved a special case (for S = the empty set) of the following theorem: If B is a smooth complex curve and S is a finite subset of B then there exist only finitely many families (up to isomorphism) of smooth curves of fixed genus g ≥ 2 over B \ S.[8] The general case (for non-empty S) of the preceding theorem was proved by Suren Arakelov in 1971.

[8][9] At the same time, Parshin gave a new proof (without an application of the Shafarevich finiteness condition) of the Mordell conjecture in function fields (already proved by Yuri Manin in 1963 and by Hans Grauert in 1965).

[7] His other research dealt with generalizations of class field theory in higher dimensions, with integrable systems, and with the history of mathematics.

[5] Parshin was an invited speaker at the 1970 International Congress of Mathematicians (ICM) with his talk titled Quelques conjectures de finitude en géométrie diophantienne.

[5][15] He was a plenary speaker at the 2010 ICM with his talk titled Representations of higher adelic groups and arithmetic.