For objects up to 10 kpc away, the triangulation and mapping of stars is
exactly how it's done:
http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit1/distances.html
The use of stellar parallaxes sets out a template of distances with extremely low levels of uncertainty for an enormous quantity of stars (the European Space Agency Gaia mission is ultimately targeting around a billion objects). This gets analysed alongside the objects' spectroscopic data (which details the star's chemical composition) and observed luminosity. The three sets of data provides a catalogue of reference data - I.e. if one observes a particular star, record its spectra and luminosity, the data can be used to estimate its distance. The level of uncertainty in such measurements isn't in the order of a factor of a million but less than 5%. The data collected is verified and/or updated through additional checks, such as planetary transits of observed stars.
Distances to nearby galaxies can also be examined using this catalogue, because individual stars can be resolved using the likes of the HST in large spiral galaxies, such as M31. This information in turn is then used to verify distances to further galaxies, along with additional reference systems found in our galaxy, such as Cepheid variable stars and supernovae - these systems use the fact that the observed events (e.g. the period of variability in a Cepheid) directly relate to the observed luminosity and if one knows how bright something should be, measured against how bright it appears to be, then its distance can be estimated (but with a large value of uncertainty compared to that achieved through parallax measurements).
One might ask how can astrophysicists be certain that the galaxy observed in this news article isn't just a very small one (which would explain its observed luminosity) that's just moving away very fast (which would explain its observed spectral redshift). The answer lies in other data - for example, the width of the spectral lines observed in the light from a galaxy
directly correlates to the size of a galaxy, so this will indicate if an observed galaxy is either small or far away. Many galactic cores, including this one, emit signals in other parts of the electromagnetic spectrum (radio, UV, X-ray, gamma) and signals at the high energy end of the scale, I.e. X-ray and gamma, lots of mass/gravity to generate and so indicate the scale of the mass of the galaxy (which in turn points to the size and luminosity).
All of this data goes together to form a picture of the scale of the distance and while individual values may suggest something closer or further away, the data set as a whole correlates to a scale of a few hundred million light years, rather than in the order of thousands or billions.