Where I grew up it was "technically" whichever parent had primary custody, which back then was usually very clear - especially during the school year. So much like taxes are "6 months and a day" for residency, it was similar for school.

In reality it was basically just "one parent lives in the district with a legal mailing address that works" - and very rarely enforced or even looked into. Especially if a kid was already enrolled and then later had a life event.

It more competitive/exclusive districts though this gets taken very seriously, with certain parents tattle-telling on others, etc.