If a council tells you that it is not using pesticides in a particular area then it is a good indication that the area in question is pesticide-free, but it is unfortunately not definitive. This is because pesticide application is conducted by varying levels of local authorities so, in some towns (for example), the town council will manage one type of land while the district or county council manages another.
Parks and green spaces are usually managed directly by the local council so if they say that they are not using pesticides in these spaces then it is usually safe to assume that these areas are indeed pesticide-free. However, it gets more complicated with areas such as roads, streets and housing estates which are more likely to fall within a council area but actually be maintained by another (usually higher) authority. In order to find out who might be spraying pesticides in your local area, it is always worth approaching your local council and asking which authority manages what land.