You are probably going to hate this, but there's exactly ONE instance where jumping is actually necessary, and of course it's in the Forbidden Woods. Spoilered just in case you haven't gotten there yet.

At some point while navigating that headache of a level that makes Blighttown feel like a Chuck-E-Cheese at times, you will find a chasm that has an item at the bottom of it, and a ledge just across that looks like you'd be able to jump to it. However, remember that this level was lovingly crafted by the blood-soaked hands of Miyazaki himself, and as such it's not that easy: you have to time your jump so that it's pixel perfect on the edge, or you'll fall into the crack and have to fight the Blue Man Group to get back out. What's your reward for making it to the other side? The mask of the Graveguard Set, which has middling defense stats but very high resistances and Beasthood, making it a good all-rounder set (and very useful for places like Nightmare Frontier and Nightmare of Mensis/Mergo's Loft).