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It's not that both my parents are opposed to my plan.
Literal
Parents [subject-が] two-people both my plan [to-に] opposing [it's-not-the-case-that-わけではない].
~わけではない hedges the negation — 'it's not the case that both are opposed,' implying one parent actually supports the plan. 二人とも (both of them) sharpens the scope, making clear what the partial negation is qualifying. A nuanced way to push back against an oversimplified assumption.