From the Kickstarter campaign:

In 1999 - 2000 range, something really weird happened: A number of people got ahold of Randy Glover, and got permission to do sequels to Jumpman. I was one of those people. Most of the agreements were to do freeware versions of the game, though, I managed to talk Randy into letting me do a commercial sequel to the game (though, I'll get into the specifics of that in a moment.)
edit: And if you read through more of it, you'll see he's got a bit of a history of trying to get this game going. I think it's safe to assume he has the appropriate rights to make this.