What would be the advantages of measuring stellar parallaxes from Mars rather than from Earth?
Measuring stellar parallax on Mars would allow the measurement of the distances of stars 1.5 times further away than can be measured on Earth.
Stellar parallax on Earth involves measuring the position of a star and then measuring the position again 6 months, or 183 days, later when the Earth is at the opposite side of its orbit. Half of he difference in angle measured in arc seconds is the stellar parallax. The inverse of the parallax angle gives the distance to the star in parsecs. One parsec is 3.26 light years.
Modern equipment can measure star positions to an accuracy of about 0.01 arc seconds. This means that parallax can be used to measure the distance to stars up to about 100 parsecs or 326 light years.
The distance the Earth moves in 6 months, or half an orbit, is 2AU. This is called the baseline. Mars is about 1.5 times further away from the Sun than the Earth giving a baseline of about 3AU. This means that stellar parallax on Mars can be used to measure the distance to stars up to about 150 parsecs or 498 light years.
As Mars takes 687 days to orbit the Sun, the two measurements would need to be taken 343 days apart rather than the 183 days apart for Earth measurements.