A row unit assembly may generally include a clamp mount configured to be coupled to a frame member of an agricultural implement. The clamp mount may include a first clamp member and a second clamp member pivotally coupled to the first clamp member. The first clamp member may include a first clamp wall spaced apart from a second clamp wall of the second clamp member such that the frame member is clamped between the first and second clamp walls. In addition, the second clamp member may include a flange extending outwardly relative to the second clamp wall such that the flange overlaps the frame member along at least a portion of a top side of the frame member. Moreover, the row unit assembly may include an agricultural row unit including at least one support arm pivotally coupled to the clamp mount.