The problem with box store softeners is they may be undersized or not set up properly also the resin now out of china really chaps my hide as it is inferior to what was available only a few years ago from US manufactures.
The white cloudiness may indeed be dissolved air. Fill a glass and from the bottom does it clear up after a few minutes? I would check the air check valve in the brine tank to ensure it's not drawing air in only to be purged during use.
Your loss of pressure from the old softener may have been a result of never having a properly set regen cycle including backwash flow rates resulting in a compressed media bed. A huge problem again with them fancy store bought softeners.
I had for years built my own using Autotrol but since GE bought them and sold them down the river to Pentair, Pentair has less than zero concern for professionals using the autotrol controls. I had switched to clack...
as the industry moved to chinese manufacturing, I had last year, several pressure vessels fail in under 75 psi conditions.
I would have a good professional look at your parents system and start with a water analysis from a state lab, before and after treatment.