These majestic mystical creatures can be tamed! They are found in the mining dimension and are really hard to tame due to their outrageous 55 attack damage (that's past a one-shot even with full armor points) and speed! However, managing it will grant you the ownership of such power.
They can be tamed with raw beef. Once tamed, right-clicking with an empty hand will mount the Leonopteryx. Sneak to dismount. You can have your dragon "sit" (that is, not follow you) by right clicking them with anything in your hand except more raw beef: that heals them, assuming they'll ever take enough damage that you'll need to..
Or, kill them for potentially a good weapon.