Yes. Casual vacancies must be advertised as soon as they occur, with the elections office usually publishing the statutory notice. This is because electors have rights to call an election, although this rarely happens and should be discouraged due to the cost. If no election is called and the council remains quorate, the elections office will usually leave the council to manage its own vacancies.
The confusion arises because for scheduled elections, Section 39(1) of the Representation of the People Act 1983 requires the district council to call a second election within 35 days of the first to fill vacancies. This is a general requirement for elections to all levels of local government. Section 21 of the Representation of the People Act 1985 (i.e. 2 years later) deals specifically with parish councils and places the responsibility within a quorate council to fill the vacancies by co-option with no external involvement. The role of the district council is limited to supporting the parish council to become quorate, which may, as a last resort, require the district council to appoint councillors to fill vacancies.