Tricky situation!
It appears that the parish council has no obligation to pay anything. Although the "contractor" might then threaten to undo the work, which wouldn't really help anyone. So it seems, as you imply, that if the parish council can afford the bill and believes that it has received good value, then it should resolve to accept the bill and pay it.
I wouldn't think this would set a precedent, but it would certainly seem advisable for the parish council to also resolve to minute its policy that it will, in general, refuse payment for unauthorised work, and could even report unauthorised work to the police as possible criminal damage. A resolution of this kind would be recorded in the minutes, and could also emphasis that financial standing orders normally apply to all purchases.
How far you would go in publicising the concerns (as well as the approval of the specific work done) is a matter to be decided in relation to what opportunities exist locally and other local factors.