The story
Paddle refers to the rider's arm action, the same as paddling a surfboard, not to a separate physical paddle (no such paddle is used with this type of board).
Etymology from Wiktionary, CC BY-SA 4.0.
paddle board
Today it means: A large fat board used for surf lifesaving. It is propelled by the rider using their arms, either lying face down or sitting up on their knees, and usually has straps positioned along the sides to hold onto. Full definition →