Yes, but there are snags.
It's unlikely that any but local advertisers would think it worth advertising on a parish or town council web site. Local advertisers may cause concerns.
The most obvious one is that the council is then treating some local businesses as customers, and this may give an impression that the council will be prejudiced in their favour.
Or it may be seen as unfair for the council to promote some businesses and not others, even though a charge is made, if the web site is at least partly paid for out of public funds.
Another complication is that so far as the web site is concerned, the council is then trading, and this may mean that the dispensation to recover VAT may be lost, at least in relation to the web site.
You'd have to judge the benefits against the drawbacks in the specific case.