I think you're comparing apples to oranges, Em. What's being discussed is not someone's hen on a one-time escape from a barnyard, the subject is a neighbor who habitually lets his chickens wander without any restraint. A rooster and hens allowed to leave the owner's property without any effort being taken to keep them in. No coop, no pen, just let the chooks out and let them go where they may. The flock habitually enters the OP's parents yard, and the broody laid her eggs in the parents' empty fountain.
Not only are the chickens endangered in this way, but the neighbor is also allowing them to be a nuisance. The rooster crows and the hens squawk under the OP's parents' windows, for pete sake. How would you feel about a neighbor letting his flock loose to come dig up your garden beds and poop on your patio? LOL!
In this case, the person may not even notice or care that one of his hens is missing. The OP hasn't mentioned the neighbor going door to door frantically trying to locate a beloved missing hen. When one of my bantam hens got away, I DID go door to door trying to find her. When she returned on her own, I made sure my coop was secure so it wouldn't happen again.
My first action would be to politely talk with the neighbor and see if some arrangement could be made to coop the flock. If he doesn't seem to care, FKLW.