I know it didn't really want an explanation but you're gonna good and gosh darn get one.

Both words predate cars.

"Parkway" used to always be literal. A way/path through a park. A wide landscaped scenic route through parks for carriages. Those scenic routes eventually became roads with the arrival of cars, often existing beyond parks and even sometimes stretching between cities, but kept the same old timey name.

And driveways weren't just the often car-length bit of pavement where your car sits. They were long private drives, roads that connected a manor home to public roadways across an estate. They were primarily for driving carriages upon between the public roads and the private manor, so the name made more sense. With the advent of cars, homes that weren't on massive estates now needed a paved or gravel path for their cars to travel to a garage nearby or attached to the home. They used the same name despite being far shorter, only needing to cross a more typical yard, requiring little driving.

And while we are on the subject a "park" as in a green space used for conservation and/or recreation and the action, to "park", as in a car, have the same origin. They both come from the French word "parc", which means "an enclosed piece of ground". The origin of the connection for that word and modern public green spaces goes back to medieval royals building fenced-in areas of wild greenery for royals and nobels to hunt in. As for what that has to do with parking a car, it goes back to a military preparations where wagons, weapons and horses would be stored and organized in designated enclosures, "parcs", thus "parking" them for later usage. When cars came about and needed organized storing, the word already existed for such a thing.