The City Council shall have control of the parkings upon each street of said City, including the right to permit, regulate, or prohibit the placing of signs within and upon said parkings. Said parkings shall be understood and held to be the space between the curb or gutter and the street line on each side of the street, except so much thereof as may be occupied by publicly owned and maintained sidewalks. The City Council shall also have control of these areas in the center of streets not used for traffic. In case said parkings are not properly cared for by the owners of the property abutting thereon, the City Council may, in its discretion, and shall, upon petition of the majority owners of the abutting property on such street or any block or blocks upon such street, cause such parkings to be properly cared for, the grass cut, weeds destroyed, and such work done as they deem necessary; and the cost of doing such work shall be assessed upon abutting property as provided in the preceding section. Provided, however, that nothing in this section shall interfere with the necessary work of the City in its work on the streets, curbing, sidewalks, or in making sewer connections in any such streets, or of the City in making or repairing water connections.