Braking chutes came in with the B-47, which actually had two chutes - a small one, deployed in the air, which allowed the pilot to keep the thrust level up on finals (otherwise, the engines wouldn't respond fast enough for a go-around, and the '47 had no airbrakes) and a big one to stop on the ground.
The 117 has a relatively high landing speed owing to 67.5 degrees sweep, no wing curvature and other wackiness, and uses the chute pretty routinely. They had black chutes when they entered service but this was abandoned because Victoria's Secret complained about them hogging worldwide supplies.
The F-16 was designed without a chute but the Norwegians insisted because they operate from icy runways - hence the Norwegian jets (and some others) have a big, boxy fairing at the tail root. The Norwegians also want a chute on the JSF and will have to pay through the nose for it when LMT figures out where to put it.
Other fighters use effective aerobraking instead of a chute - big speedbrakes, differential control surface actuation &c.
The 'phoon has a chute option, I think, because of operator insistence/wet-icy runways &c. But it also has big fan-cooled wheelbrakes (don't have to wait as long for brakes to cool between sorties) an airbrake and the canards, which can be used to dump the aircraft on the wheelbrakes.
By the way, early Caravelle airliners had braking chutes.