I continue to exchange letters with my parents.

Literal

Parents [with-とは] letters [possessive-の] exchange [object-を] am-continuing.

手紙のやりとり (exchange of letters) uses やりとり — a noun meaning 'back-and-forth exchange.' 続けています (am continuing) is polite ている. とは combines the conjunctive と (with) and は (topic/contrast).