Some councils used to have a condition that (in order to avoid a penalty for incorrect parking) a valid tax disc needed to be displayed on any vehicles on their pay-&-display car parks. However a court has ruled such a condition to be unlawful, so it would seem unlikely that the car park operator could take any action in respect of an out-of-date tax disc. (While publicly accessible car parks are regarded as 'public places' for the purposes of some motoring legislation - such as 'dangerous driving' offences - they are regarded as 'private property' in regard to the display of tax discs).
However if you've got your new tax disc before you go, you can display it as soon as it's in your possession. Tax discs don't have a 'start' date on them; they only declare that they're valid up until a specific 'end' date. So, for example, if your tax expires at the end of this month (and you've received the renewal form) you can go out and buy a new tax disc today and put that disc onto your vehicle straight away. You don't have to wait until the end of the month to do so.
Chris