The construction industry lacks solutions for fast, accurate and comprehensive tracking of the 3D status of building elements. Such information is however critical to the successful management of construction projects, as it can support multiple fundamental activities, starting with construction dimensional quality control. An approach for automated extraction from construction laser scans of point clouds corresponding to the elements constituting project 3D CAD models has been initiated and previously presented. This novel approach, which is still being improved, enables the construction of Project 4D Information Models (P4dIMs) which then support the management activities stated above. Approaches for using P4dIMs for supporting such activities have however been sketched, but not implemented and tested. In this paper, the author presents preliminary work in the use of P4dIMs and parametric object fitting algorithms for automatically inferring construction dimensional quality information. For any given project 3D element, parametric objects are fitted to corresponding point clouds extracted from the P4dIM. The parameters of the fitted form can then be compared to those of the designed one with respect to construction specifications. Flags are raised for further investigation when deviations are observed. The challenges of this work lie mainly in fitting parametric objects to incomplete object point clouds.