Using snapshots from that film, which is a head-on view of the ship in which the starboard, or right side is on the left of the image, I labeled the major parts mentioned in this post.
This diagram may help explain some risks that salvors off Giglio will be managing tomorrow (if the weather holds), when they'll have to keep a close eye on concentrations of stress in the hull as it rotates under pull from the parbuckling cables and later the weight of the sponsons. The salvors have been pretty clear that things will bend and break during the process ... hopefully not the bigger things.
I'm guessing one risk they'll be watching will be a temporary concentration of weight on the starboard edge of the hull (see yellow arrow, below). The concern is weight of steel not fully offset by displacement in water: that's hull, fittings and the new sponsons, which will be mostly out of the water at that point.
Ship designers sometimes call this area of the hull, where horizontal plating meets vertical plating, the "turn of the bilges." It's the location of a stabilizing fin called the bilge strake, depending on vessel details. Here's a diagram from bestshippingnews.com -- see the highlighted yellow portion.
Recall that no seagoing cranes will be attempting to hold the weight of the hull off the seafloor. That's the nature of parbuckling: the structure has to hold up while a lateral, rolling force is applied.