Parallax versus Redshift when measuring astronomical distances? From my understanding, the most common method for calculating how far away stars and galaxies are relative to us is by accounting for their redshift. I feel that this method could be inaccurate, though, as interference from dust clouds, neighboring galaxies, relative speed of the object, gravitational lensing, etc, might interfere with how shifted the light that reaches us really is.
However, I feel that such issues wouldn't be as prominent with parallax. If we had two satellites at a considerable distance from each other measuring the same patch of the sky at the same angle, or if we take two measurements of the observed objects from Earth (or its orbit) within a 6 months interval, we would be able to tell more accurately how far that object really is. The margin of error would be defined by how precisely the instruments are lined up, rather than several different hypothetical factors from unseen sources.
With that said, I feel like that, the further the object is, the harder it'd be to pinpoint its distance to us with considerable accuracy. We'd have to rely on the smallest of scales of measurements, which would still give astronomical margins of error. But couldn't the same be said about redshifting?
My question is: Is parallax measuring more accurate than redshift measuring when it comes to distance?
And, if I might tie in a follow-up question, in case parallax is more accurate, why haven't there been as many big projects to explore that method as redshift observations?
 A: The realms of applicability for redshift (as an indirect distance indicator, that must be calibrated through the Hubble parameter) and parallax, which can only be applied to relatively nearby stars, do not overlap.
Parallax is a geometric method, limited only by the precision with which the parallax can be measured. At present it is limited to distances of hundreds of light years, though the Gaia astrometry satellite will soon extend this to thousands of light years. 
Redshift as a (reliable) distance indicator can only be used well into the "Hubble flow", so that individual Galactic peculiar velocities become unimportant compared with the universal expansion. In practice this means tens of millions of lightyears at least. The technique cannot be used at all on local group galaxies or stars in our galaxy, since their motions are not due to the expansion of the universe.
You are correct that parallax is more accurate, but current levels of measurement precision are too low to extend the technique to distant galaxies.
One way of looking at this is to ask what parallax baseline would be needed to measure a parallax at 100 million light years with current measurement technology.
Gaia can measure stellar positions to about 10 microarcseconds. The parallax at 100 Mlyr is about $0.033b$ microarcseconds, where $b$ is the parallax baseline in astronomical units (the Earth-Sun distance). To get a parallax measurement precise to 10% would require a baseline of $b \sim 3000$ au - i.e. larger than the solar system.
