Proceedings to establish local improvement districts must be initiated by petition in the following cases:
(1) Any local improvement payable in whole or in part by special assessments which includes a charge for the cost and expense of operation and maintenance of escalators or moving sidewalks shall be initiated only upon a petition signed by the owners of two-thirds of the lineal frontage upon the improvement to be made and two-thirds of the area within the limits of the proposed improvement district;
(2) If the management of park drives, parkways, and boulevards of a city has been vested in a board of park commissioners or similar authority: PROVIDED, That the proceedings may be initiated by a resolution, if the ordinance is passed at the request of the park board or similar authority therefor specifying the particular drives, parkways, or boulevards, or portions thereof to be improved and the nature of the improvement.