Gaia's five-year observation baseline might naively lead to the expectation that it will be possible to fit the parallax of any sufficiently nearby object with the default five-parameter model (position at a reference epoch, parallax and proper motion). However, simulated Gaia observations of a ‘model Universe’ composed of nearly 107 objects, 50% of which turn out to be multiple stars, show that the single-star hypothesis can severely affect parallax estimation and that more sophisticated models must be adopted. In principle, screening these spurious single-star solutions is rather straightforward, for example by evaluating the quality of the fits. However, the simulated Gaia observations also reveal that some seemingly acceptable single-star solutions can nonetheless lead to erroneous distances. These solutions turn out to be binaries with an orbital period close to one year. Without auxiliary (e.g., spectroscopic) data, they will remain unnoticed.