For example, "The tin-roofed cabin near Lake Nicotoon was iconic: It was built by the parents of Tom and Archie when they first moved to Florida in 1938."
Is the bolded sentence differ from "The tin-roofed cabin near Lake Nicotoon was iconic: It was built Tom and Archie's parents when they first moved to Florida in 1938"?
In almost all situations, the "of" form and the apostrophe (') form have the same meaning. Both syntaxes are used for the "genitive" concept in English, which means either owning, possessing, or being part of something.
Archie's father (the father of Archie)
the building's front door (the front door of the building)