You're at risk of overthinking this! Firstly, S101 allows councils to cooperate with each other, as willing partners. It doesn't allow them to dump their responsibilities on another council. If it did, would any council ever spend any money?
Highways maintainable at public expense are sometimes referred to as the Queen's (now King's) Highway and this is the collective term for all types of highway adopted by the highways authority, either specifically or by default. The highways authority is the funder of first resort and has many legal duties enshrined in legislation. Funding may come from a variety of sources, including the Council Tax, government grants, utility companies, contributions from developers or landowners, EU funding (pre-Brexit), etc, but it remains the responsibility of the highway authority to undertake its statutory function to deliver. Parish councils can and do contribute in a small way to some schemes, but the only specific powers to do so are for the maintenance of footpaths and the provision of traffic calming schemes. Several of my councils have made modest contributions to other highway improvement schemes delivered by our county council.