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Gravitational-wave astronomy allows us to study objects and events invisible to electromagnetic waves. So far, only signals triggered by coalescing binaries have been detected. However, as the interferometers' sensitivities improve over time, we expect to observe weaker signals in the future, e.g. emission of continuous gravitational waves from spinning, isolated neutron stars. Parallax is a well-known method, widely used in electromagnetic astronomical observations, to estimate the distance to a source. In this work, we consider the application of the parallax method to gravitational-wave searches and explore possible distance estimation errors. We show that detection of parallax in the signal from a spinning down source can constrain the neutron star moment of inertia. For instance, we found that the relative error of the moment of inertia estimation is smaller than $10\%$ for all sources closer than 300 pc, for the assumed birth frequency of 700 Hz, ellipticity $\geq 10^{-7}$ and for two years of observations by the Einstein Telescope, assuming spin down due purely to quadrupolar gravitational radiation.