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In principle yes but an orbit round the Moon is too small to make it worth doing. An orbit round Mars would also be too small.

The present system uses the Earth's orbit as the base line, and parallax measurement works by measuring the exact position of a nearby star agains the background of distant stars at intervals of 6 months at opposite sides of the Earth's orbit.

The parallaxes are so small that it took until the 1800s for any parallax to be discovered. Before then the lack of parallax was always used as a genuine reason that the Earth could not be moving.

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