So I'm writing a thing about the adventures of a guy living in an alternate history version of Earth in the 15th century, where a genius invents hot air balloons in the 14th century, along with workable parachutes. What I'm wondering is, how could the protagonist accidentally invent the hang glider? Or any other sort of gliding thing made to work with one or more passengers?
Or, to follow along with what you have given us, perhaps the protagonist finds a need to use his parachute and during his descent grabs hold of the lines in an effort to steer and finds himself accelerating forward, maybe because one of the lines is tangled creating a different shape. This might give him the idea to play with the shape of the parachute leading to a more modern design and then through experimentation to the wing.
It doesn't have to be too terribly accidental. There have been no shortage of attempts at flying in history; they generally failed because the machines were either too heavy or based too strongly on birds.
Your inventor could discover the basis of a practical hang glider when the flapping mechanism of their latest attempt at bird-like flight jammed just after launch, locking the wings in the most advantageous position. Efforts to un-jam it could result in the accidental discovery of weight-shift control.
Anything is possible
Imagine your protagonist is taking the advantage of the high winds on the edge of a steep hill, cliff, or canyon to dry a sheet or other large piece of cloth. Your protagonist doesn't want his sheet to get blown around too much, so he uses wooden posts to tie down his sheet, stretching it taught over a series of wooden posts.
Then an intense scene shows your protagonist to accidentally push the sheet-dryer off the edge of the hill. While she won't fly very far, she will feel the pressure of the wind and possible experience some hang gliding power. This will allow him to later explore and develop the idea into a more feasible hang glider.
See Leonardo da Vinci's glider for some ideas for a very basic glider design.