The third data release of Gaia, in June 2022, included the first large sample of sparse photometric data for more than 150000 Solar System objects (SSOs), mainly asteroids. The SSO photometric data can be processed to derive information on the physical properties for a large number of objects, including spin properties, surface photometric behaviour in a variety of illumination conditions, and overall shape. After selecting a set of 22815 objects for which an adequate number of accurate photometric measurements had been obtained by Gaia, we applied the "genetic" algorithm of photometric inversion developed by the Gaia Data Processing and Analysis Consortium to process SSO photometric data. Given the need to minimise the required data processing time, the algorithm was set to adopt a simple triaxial ellipsoid shape model. Our results show that in spite of the limited variety of observing circumstances and the limited numbers of measurements per object at present (in the majority of cases no greater than 40 and still far from the number expected at the end of the mission of about 60-70), the proportion of correct determinations for the spin period among the observed targets is about 85%. This percentage is based on a comparison with reliable literature data following a moderate filtering procedure developed to remove dubious solutions. The analysis performed in this paper is important in the context of developing further improvements to the adopted data reduction procedure. This includes the possible development of better solution filtering procedures that take into account, for each object, the possible presence of multiple, equivalent spin period solutions that have not been systematically investigated in this preliminary application.