These are in use because they are "natural" distance scales in astronomy.
An AU (astronomical unit) is the mean distance from the Earth to the Sun, which is a very useful length for comparing distances within the Solar system (or within others!), not to mention very relevant for research that involves the Earth's motion.
A light year is the distance that light travels in a year. Since light travel time is exactly the time lag in a signal (such as an astronomical observation), it makes sense to use "light-years" - If a thing is 10 light-years away, we are seeing it as it was 10 years ago.
A parsec is a little more than 3 light-years, and is derived from the the term "parallax-second", or an apparent motion of a star of one second of arc (1/3600th of a degree) over the course of an Earth-year. While the parsec has technically been superseded by the light-year in many ways, it does seem to be a more natural distance with in the Galaxy; the nearest star is 4.2 light-years away, but only 1.3 parsecs. Galactic astronomers (such as MilkyWay@home researchers) traditionally use "kiloparsecs" (thousands of parsecs) as their chosen distance scale.
With any unit, there is a trade-off between mathematical simplicity and human relevance. A mile was originally defined as a thousand paces, with a pace being ~5.28 feet, or the distance an average human covers in two steps. A pace is also close to the average height of a human. Personally, I don't feel like walking a kilometer is a very far distance, but a mile feels significant. I kind of wish that we were using paces and miles instead of meters and kilometers...