IMO light-seconds is actually a really nice unit for interplanetary distances. 1AU ~= 500ls. Not only is the scale nice for human arithmetic, but it serves a dual purpose of also telling you the light delay.
I'm pretty interested in this kind of thing but I couldn't tell you to within an order of magnitude Earth's distance from the Sun in metres or miles. I could work it out - it's about 6 light minutes and light travels 300,000 km/s - but I wouldn't expect everyone to do that. So expressing the distance in AU is relatable to me. Looking from the probe at perihelion, the sun will appear 25 times wider, or 625 times larger, than it does from Earth.
And does the fact that AU comes from the earth's distance to the sun make it self-centred? Sure, but everyone we need to communicate with would equally find it centred on them. Compared to other measures like "the distance from the king of England's nose to the tip of his outstretched finger" it's quite neutral.
If one AU is 1 meter, 0.04 AU is 4 cm. So it's, roughly, if 3 feet is between the Earth and the Sun, the probe would pass by the Sun at one and half inch.
(It is obviously easier to think in metric units.)
> If Earth was at one end of a yard-stick and the Sun on the other, Parker Solar Probe will make it to within four inches of the solar surface
Perhaps they meant 4 cm.