Continuity mistake: When Lane unpacks the groceries Zack has brought, she takes out the bananas and then the apples, twice in a row.
Continuity mistake: In this episode, when Logan goes to see Lorelai to ask her permission to propose to Rory, Lorelai asks Logan if he wants any french fries. They sit down and the to-go container is open, when the camera zooms in and then back out it's closed again.
Answer: Yes, Lane lives with both her parents. Her father does live with them, but he was only seen in the Netflix revival.