In a multi-part cover for a storage space of a roof in the rear of a vehicle, which cover comprises a center part and side parts pivotally connected to the center part, a common operating mechanism for operating the parts of the cover depending on the position of the roof is provided with a coupling between a roof operating linkage and a cover operating linkage whereby the roof operating linkage can be uncoupled from the cover operating linkage when the roof is deposited in the rear storage space for permitting longitudinal movement of the cover into an end position, in which it covers the roof in the storage space and the side parts of the cover are pivoted outwardly from the center part of the cover.