At any time following the formation of a district under this part and the levy of the initial assessment in such district, any parcel of land, whether contiguous or noncontiguous to the district, may be added to such district if the owner of such parcel files with the legislative body a petition for such inclusion and the legislative body grants such petition. The legislative body shall not grant such petition until the petition has been approved by the board of parking place commissioners of the district and until the street superintendent has estimated the amount which such parcel would have been assessed and taxed under all prior assessments and special ad valorem tax levies in the district if said parcel had been a part of said district when formed, and until the owner of such parcel has paid to the city such estimated amount. All such amounts paid to the city shall be placed in such fund as the board directs and may be used for any purpose for which revenues from fees, charges and rentals for the use of parking places of the district may be used. If such petition is granted, such parcel shall thereafter be entitled to the same rights as those which apply to other lands in the district and shall be subject to all assessments and special ad valorem taxes thereafter levied in said district under this part.