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She wouldn't let the children go out in the rain.
Literal
She [topic-は] children [subject-が] rain [middle of-の中を] go-out [nominalizer-の] [object-を] did-not-permit.
雨の中 ('the middle of the rain, in the rain') is the standard way to describe an action performed during rainfall — a noun-phrase setting rather than the prepositional 'in the rain.' The を after 雨の中 marks it as the path traversed (a less common use of を with motion verbs). 許す is the verb for permitting, allowing, or forgiving; in the negative 許さない here, it's a parental refusal to let the action happen. 出て行く ('go out, leave') combines 出る + 行く to capture both the leaving motion and the moving-away direction.