Here's the order of increasing length for the astronomical units you provided, from shortest to longest:

1. Kilometer (km): The smallest unit, representing 1,000 meters.

2. Astronomical Unit (AU): The average distance between the Earth and the Sun, approximately 150 million kilometers.

3. Light-Year (ly): The distance light travels in one year, approximately 9.46 trillion kilometers.

4. Parsec (pc): A unit based on parallax, approximately 3.26 light-years, or 31 trillion kilometers.

So, the complete order is: kilometer < astronomical unit < light-year < parsec.