Which is correct parent's details or parents details?

Single parent-> parent's
Multiple parents-> parents'

The only instance I can imagine when "parents details" would be correct, when the details apply not to the parents themselves but to something else of value or interest to a parent or connected to them in some insignificant way. For example, Jobs, a cadre of parents visits a museum, then weeks later a class of children. In discussing these two visits we consider the trash left behind by each group considered as a detail of their visit: Thus parents details vs kids details. Is it a stretch...

You bet it is. However in this case my Better Half, who is much smarter than I, agrees

I'm with Fresco on this; the topic is singular and plural possessives:

A (or the) parent's details: details of a (or the) parent.
Parents' details: details of the parents.

Very comprehensive summary:

http://alt-usage-english.org/excerpts/fxwheret.html