For 1958, Ford confounded me (totally). They had a change in March which affected so many models. The confusing part is, the same body-code number didn't change, which meant the same car could look totally different for the same year, and the patent plate became much more important when ordering parts.
The easiest way I can identify these models is by trim. If you would like to see the different models, CLICK HERE.
I excluded Retractables, T-birds, Ranchero's, Sedan Delivery's and wagons, and still came up with eighteen different cars.
DKheld, to answer your question directly, your Mom and Dad's car body matches a 1958 Fairlane Town Sedan (58A, built after March 03, 1958).
Remember, Galaxie didn't debut until 1959. Your parents had a real nice car. I'd love to have one. BTW, Ford was installing 292 Y-Blocks in those cars, as well as FE engines. My '59 Galaxie came with a 292/3-speed stick on the column.