She is wrong.
Any individual or group can campaign in a referendum.
They can spend £2362 + (0.059p per elector). Records of spending must be made and returned to the district council in the same way as an election expense.
Whilst the parish council itself SHOULD NOT be an active campaigner in the referendum (code s4(1) LGA 1986), they can state whether they support it or not.
But the councillors can, as a individuals outside of the parish council structure, run a campaign for or against.