I am not sure how detailed an answer you require, but I will focus on your most specific question: how do we read a red shift?
The answer lies in the familiar. We know that a normal spectrum will have "absorption lines": black lines where a specific frequency (usually more than one of them) of light is absorbed by a specific element. Each element has its own, distinct pattern of absorption lines, so we can seek them out and see how far they have shifted from their original position in the spectrum. Dividing the amount of shift by the emission (when working with wavelengths) will give the red shift when working with wavelengths, and a similar approach works with frequencies (I am oversimplifying since I am not sure how detailed you want this). If you get a negative score, it means the object is blue-shifted and approaching us, whereas positive means it is red-shifted and moving further away. Astronomers have shown that galaxies roughly 1,000 megaparsecs away (or less) will have a shift correlating closely yo the galaxy's actual distance.
You are correct that using parsecs requires a relatively moderate distance, since you need to measure the apparent difference in angle of the object from two vantage points. Without starships, we can only maximize this by observing the object from either end of our orbit. Distant objects are so far away that there is no accurate measurement of their difference in angle.
