In 2 parts,
Yes a Parish Council can buy any equipment necessary to fulfil a legal duty, contract or obligation.
The next part, is what is the contract / duty / relationship to the private land? If there any type of relationship, e.g. a grant of access or lease with requirement to maintain the land, then yes it is perfectly legal.
But they can't just mow Mrs Smith's lawn because they want to, unless ... they employe a mowerman to do the verges and to help raise revenue they have a trading business that the mowerman goes and cuts people's lawn for revenue. (An unlikely, but perfectly legal scenario).