I don't have experience of this happening, but would expect you to be very much at risk of neglect and/or overcharging. It is unlikely that a District Council could manage parish affairs efficiently or with the same degree of care one might expect of local people. At one time you might have expected more of a goodwill effort, but with financial pressures and staff cuts, most councils now struggle to meet their own commitments.
Keeping up pressure on the parish council would be my preferred route. Force parish meetings and pass critical motions, seek press publicity, line up some alternative candidates for the council. Circulate a newsletter identifying the problems. With continuing efforts and public disapproval it is quite possible that the unsatisfactory councillors will eventually throw in the towel and resign. Good luck!