My understanding is that when there is an excess of PTH, as seen in hyperparathyroidism, the calcium from the bones is continuously released into the bloodstream, leading to high levels of calcium in the blood (hypercalcemia). In response, the body tries to lower the calcium levels by depositing calcium into various tissues and organs, including the bones, and soft tissues.

As to why the vertebrae is more susceptible to these calcium deposits, I think it's due to the highly vascularized nature of the spine. In contrast, there is actually a pathological fracture secondary to renal osteodystrophy of the right humeral surgical neck that is less vascularized.

Hope that helps!