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We can't use parallax to measure a stars distance from the Earth if the star is already too far away. The angles used in parallax measurment are already very small, and if the star is beyond a certain distance from us the angle becomes too small to measure, and no distance can be determined.To date the largest distance that can be measured using parallax, with the Hipparchos sattelite, is about 1 600 light years. This will be improved with the European Space Agencies Gaia mission in 2012 and 2013.
At farther distances, the parallax becomes too small to measure accurately. At a distance of 1 parsec, a star would have a parallax of 1 second (1/3600 of a degree). (The closest star, Toliman, is a little farther than that.) At a distance of 100 parsecs, the parallax is only 1/100 of a second.
It means that the distance is greater than a certain amount - depending on how precisely you can measure the parallax.
If a star's parallax is too small to measure, it means that the star is far from Earth. Parallax measurements are used to determine the distance of nearby stars by observing their apparent shift in position as Earth orbits the Sun. Stars with large parallaxes are closer to Earth, while stars with small or undetectable parallaxes are further away.
It means that the distance is greater than a certain amount - depending on how precisely you can measure the parallax.
The farther the object, the smaller its parallax. In this case, the parallax is about 1/300,000 of an arc-second (and an arc-second is 1/3600 of a degree) - way too small to measure. Perhaps you will eventually find a way to measure smaller parallax angles.
The parallax angle of such distant objects is way too small to be measured. In general, the farther away an object, the smaller is its parallax angle.
You can conclude that it is farther than a certain distance. How much this distance is depends, of course, on how accurately the parallax angle can be measured.
At larger distance, the parallax becomes smaller, and therefore harder to measure. Even the closest star (Toliman) has a parallax of less than one arc-second (1/3600 of a degree), which is difficult to measure. Stars that are farther away have a much smaller parallax.
On the contrary, if the parallax angle is too small, it can't be measured accurately.
It means that its distance is farther than can be detected. For example, if the smallest angle that can be detected is 1/100 of an arc-second, it would mean that the star is farther than about 100 parsec.
Did you ever sit in the passenger seat and look at the fuel gauge on the dash? You see the gauge from the side so it appears that the needle is pointing to Empty. The driver is looking straight at it so the driver sees the actual reading to be a quarter of a tank. That was a parallax error that your observation position created.