。
My parents absolutely refused to let me date men.
Literal
Parents [topic-は] I [subject-が] men [with-と] associate [nominalizer-こと] [object-を] absolutely permit-try-to [contrastive-は] didn't.
許そうとはしなかった is a strong negative: 許す (permit) + ようとする (try to) + はしなかった (didn't even attempt to) = 'absolutely refused to even consider permitting.' 絶対に reinforces total refusal. 付き合う means 'to go out with/date.'