Yes, KY court records are sorta limited. Ya gotta really dig.

The son may have asked his parents if a friend could park the vehicle there. Son may not live with them. If he does, then it's possible that the home they are saying she walked away from, is the other guy's home. Most folks would notice that their son strolls in at 4 a.m. with a stranger in tow. Then again, his parents may have left him to house-sit, and later, LEOs come knocking, with a search warrant in hand. That particular residence's boundaries are outlined in red, in the google shot, so it's not a large lot (about an acre) and should have been fairly easy to search.
If this is the same vehicle, as in the links/photos below, though, then I've a feeling it belongs to the son, or someone else who either lives at, or frequents, that property;
Starting at 2:03 in vid at link, I think that is the vehicle, that they load onto the rollback.
Police Search Home Tied To Man Last Seen With Missing Richmond Woman
The below pics show the vehicle on a rollback, (Vehicle Seized) and, I think..., that may be the same vehicle that you can see in the Google overhead shot, too (Car Site).
If so, that vehicle has a history of being there since at least 04/2017.