If the parish council will manage the campaign, they may accept donations from members of the public towards the costs incurred. This is no different from a public appeal for something like a defibrillator. The whole process, from the initial deliberations of the parish council over their involvement, to their agreement to accept donations, to the control and use of the money, the involvement of members of the public etc, must be conducted in accordance with the council's standing orders and financial regulations, as if this were a normal parish council project with no public involvement.
Parish councils have powers to hold endowment funds and manage charities on behalf of others, so there may be no restriction on holding third party funds, although I can't find a legislative proof of this. In these cases too, the money must be treated in the same way as the council's own funds and reported accordingly.