As regards on-street, 'No overnight parking' is not a permitted expression neither is it a permitted variant of a prescribed sign.
https://www.legislation.gov.uk/uksi/2016/362/schedule/18/made
Be careful because it's not just private car parks which operate under Ts and Cs, so do those provided by councils.
Presumably the council signs in Bedfordshire are there to stop lorries parking overnight? The parking prohibition signs on the A20 in Kent are time-specific, the >7.5T yellow box with an addition sign 18.40-2400, and 0001-0630.