I note that you've skipped over para 3! Section 137 applies to all tiers of government, not just parish councils. The power for parish councils is defined in paragraph (3) thus:-
"A local authority may, subject, in the case of a parish or community council, to the following provisions of this section, incur expenditure on contributions to any of the following funds, that is to say—
(a) the funds of any charitable body in furtherance of its work in the United Kingdom; or
(b) the funds of any body which provides any public service (whether to the public as a whole or to any section of it) in the United Kingdom otherwise than for the purposes of gain; or
(c) any fund which is raised in connection with a particular event directly affecting persons resident in the United Kingdom on behalf of whom a public appeal for contributions has been made by the Lord Mayor of London or the chairman of a principal council or by a committee of which the Lord Mayor of London or the chairman of a principal council is a member or by such a person or body as is referred to in section 83(3)(c) of the Local Government (Scotland) Act 1973."
So charitable, not-for-profit, or public appeal of a very limited and specific nature. My inclination would be that a joint project with a neighbouring council (which evidently provides public services otherwise than for gain) would be acceptable in this regard, however the need for the benefit to be commensurate with the sum expended would be more challenging in the case of a planning appeal, as there is no benefit from losing.