Wouldn't a parachute work? I know very little about flying/aerodynamics, so I might be missing something obvious.

It would work in some cases, but if you're too low it won't necessarily have time to slow you down enough. The record for the lowest BASE jump is about 100 ft, and that wasn't a surprise equipment failure. Parachutes would help in a lot of situations, but you would still easily die in a lot of other situations.

Hm I was thinking that, but that would be just as true for airplanes and paramotors going at high horizontal speeds at low altitude.

Or perhaps the issue is that, given the lack of protection in a jetpack, the lowest altitude at which you can be hurt from a fall is much lower?

I believe that's part of it. A jetpack gives you a lot less protection and the near guarantee of an uncontrolled fall if it fails. Airplanes with engine failures can still attempt to glide to a landing.

Ballistic parachutes exist for light aircraft. Seems like they'd work here as well.

https://en.wikipedia.org/wiki/Ballistic_parachute