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It's not that both my parents are strict with me.
Literal
Parents [subject-が] two-people-both me [to-に] strict [it's-not-the-case-that-わけではない].
~わけではない is partial negation — 'it's not the case that both are strict,' implying one parent is actually quite lenient. 厳しい (strict/harsh) takes に for the person being treated strictly. 二人とも (both of them) sharpens the scope of what's being qualified.