The AU and parsec, the two most commonly used astronomical distance measures, come from the fact that they depend on the most easily-measured things in the Universe: the parameters of Earth's orbit itself.

The AU is the distance from the Earth to the Sun. Geometry can easily tell us that, say, Venus is about two-thirds as far from the Sun as we are, or that Mars is about half again as far, but it can't actually tell us the Earth-Sun distance directly without some other pieces of information. Thus, a lot of the details of the solar system got worked out using the Earth-Sun distance (that is, 1 AU) as a baseline; as our estimates of the AU got better, so too did our estimates of other things.

Similarly, the parsec also depends on the Earth's orbit. In that sense, the AU and the parsec are closely related. Specifically, the parsec is the distance at which an object's position in the sky changes by 1 second of arc (1 / 3600 of a degree) over a distance of 1 AU. Mathematically, that means it's 1 AU times cotangent(1/3600 degree) = 1 AU times ~206,264, which works out to a little over 3 light-years. We use the parsec because measuring these angles is how we first established how far away the stars are, which let us develop systems for figuring out the distance to more distant stars.

Today, the parameters of the Earth's orbit are known to very high precision in terms of things like the kilometer, but that wasn't always true. And having probes far enough out to have meaningful light-travel delays is even newer.

Today, the AU is defined in terms of the meter, and the meter is defined in terms of light travel, so in a sense we actually do measure with light travel times. We just do it in a weird way for historical reasons.