Luckily, the bottom rung of the distance ladder, parallax, is only dependent on geometry, and is therefore completely solid. With the Gaia space telescope, there are now parallax measurements almost to the center of the Milky Way.
Having parallax measurements to that distance helps to build multiple versions of the next rung in the ladder (such as Cepheids and the tip of the red giant branch).
i think one of the things astronomers were pleasantly surprised to see was that the Gaia results have indicated that other low rungs of the distance ladder were pretty well calibrated. Most of these are based in some way on stellar evolution, and given that stars can be kind of messy there was always a bit of nervousness that maybe our systematic errors are larger than we think. But the models turned out to be pretty good.
parallax is only perfectly solid before GR. once you accept that space can bend it becomes a lot more complicated (especially given that we currently think ~75% of mass is dark matter which could be bending light without being visible
General Relativistic effects are taken into account by Gaia, but they're dominated by Solar System objects. Space is very close to flat, unless you're close to a very massive object.
Having parallax measurements to that distance helps to build multiple versions of the next rung in the ladder (such as Cepheids and the tip of the red giant branch).