Any city or town contiguous to or surrounded by a weed district formed under this chapter shall provide for the destruction, prevention and extermination of all weeds specified in the petition which are within the boundaries of such city or town, in the same manner and to the same extent as is provided for in such surrounding or contiguous weed district; and it shall be the duty of those in charge of school grounds, playgrounds, cemeteries, parks, or any lands of a public or quasi public nature when such lands shall be contiguous to, or within any weed district, to see that all weeds specified in the petition for the creation of such district are destroyed, prevented and exterminated in accordance with the rules and requirements of such district.