I think the one listed as 1/35th Aviation is mismarked. The Institute of Heraldry lists it as the 1st Battalion, 58th Aviation Regiment - and even then, it's only for selected elements, since the entire 1st Battalion isn't on jump status.
HHC, 1-58th Avn is based at Fort Bragg, NC, as a unit of the 18th Abn Corps.
Co A supports the 18th Abn Corps.
Co B supports the 82nd Abn Div at Fort Bragg.
Co C supports the 101st Abn Div (AAslt) at Fort Campbell.
Co D supports the 3rd Inf Div (Mech) at Fort Stewart.
Co E supports the 10th Mtn Div (LI) at Fort Drum.
Co F supports the 1st Cav Div at Fort Hood.