Where there are vacancies because too few candidates were nominated, and sufficient councillors have been nominated to make a quorum, then the council should co-opt additional members to make up the numbers. When a casual vacancy occurs the council must declare the vacancy within a reasonable time. In the following 14 days, ten electors of the parish can demand a poll. Otherwise, the vacancy should be filled by co-option. That is the 14 day rule, as you indicate.
So, once the 14 day period has elapsed after the declaration of a vacancy, there is no option to demand a poll. However, the council should be taking steps to find suitable candidates for co-option. If it is not acting reasonably, you could raise the matter with the Monitoring Officer, who is probably the senior legal officer of the district council. They may wish to exert pressure on the council to take steps to fill the vacancies.
Wanting a poll does imply that there are available candidates. If that is the case, they could put themselves forward for co-option, which might encourage the council to get moving!