As long as the council as update, the Clerk can follow the councils usual procedure (even if that is in advance of the May meeting) to seek expression s of interest / applications for consideration as per the council's policy.
If the Parish council is not update, the district council will appoint such councillors as it sees fit.
Whilst the procedure (putting up adverts/notices) may start under the old council, the old council must not consider applications, but leave it for the new council. (Even if, in general, they are the same councillors).