Parent tube is the uncut tube. Horizontal doesn't matter. You need the angular relationship between the two tubes. Most frame design programs will give you those angles.(I use RattleCad) Offset is the distance from the centerline of the tube and will provide an off center miter. Most tube miters have a zero offset. Offset is helpful for things like moving the seat tube forward on the bottom bracket to provide more tire clearance. I also offset miter my chainstays to make them lower on the bottom bracket for ease of welding and to prevent them from contacting the seat tube.