Is there any way Invoices can be moved from a Section to another Section (or to the Parent) within a Job?

It's not possible to move invoices that way. Once a payment is applied and reconciled, that transaction is locked in the accounting package.

The only way to rectify that is to backtrack manually by going into your accounting package and undo the payment, unreconcile, and then cancel the invoice in Ascora and then recreate the invoice in the correct section and reconcile it again in the accounting package.