They Shoot Gilmores, Don't They? - S3-E7
Visible crew/equipment: Before the dance marathon, when Lorelai and Rory are headed over to the first aid station they meet up with Sookie, and when Lorelai and Sookie cross the street we can see the T-marks taped to the sidewalk. (00:15:15)
They Shoot Gilmores, Don't They? - S3-E7
Visible crew/equipment: After Luke fixes Lorelai's broken heel it cuts to Lane at the first-aid/rest stop at Miss Patty's, and when Lane walks toward her table, we can see the T-marks taped to the floor. (00:35:10)
Answer: Yes, Lane lives with both her parents. Her father does live with them, but he was only seen in the Netflix revival.