The original definition of a parking place referred to the place where trees and shrubbery were planted. When automobiles became popular, the term parkway referred to a driveable thoroughfare that ran in between said parking places. Thus the reason we drive on parkways.
A driveway is defined as “a private carriageway for a motor vehicle alongside, in front of, or leading to a house, garage, or other building; a drive”. While the definition is much the same, the lengths of these “private carriageways” have significantly become shorter with suburban expansion. Thus the reason we park on driveways.
As it turns out, there’s a perfectly valid explanation for why we drive on parkways and park on driveways- the majority of words have been around longer than cars and many have seen their definitions evolve over time, as parkways and driveways have likewise evolved.