GMAT Sentence Correction (SC) | Though initially opposed to the measure, the governor

It depends whether you consider "his" outside or inside the parallel structure. If considered outside, using once is alright. But if considered, inside it has to be repeated. Since withX and Y construction one can decide where the parallel structure begins, both are correct.
His (X and Y)... correct
(His X) and (his Y)---
...

---