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?