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The Gaia satellite will soon probe parallax on cosmological distances. Using the covariant formalism and considering the angle between a pair of sources, we find parallax for both spacelike and timelike separation between observation points. Our analysis includes both intrinsic parallax and parallax due to observer motion. We propose a consistency condition that tests the FRW metric using the parallax distance and the angular diameter distance. This test is purely kinematic and relies only on geometrical optics, it is independent of matter content and its relation to the spacetime geometry. We study perturbations around the FRW model, and find that they should be taken into account when analysing observations to determine the parallax distance.
Syksy Räsänen (Mon,) studied this question.