A marine drive is provided. The marine drive includes a propulsion unit, a supporting cradle that couples the propulsion unit to a transom bracket for attachment to a marine vessel, and a cowling system that at least partially covers a portion of the propulsion unit and a portion of the supporting cradle. The cowling system includes multiple cowl components, and at least one of the multiple cowl components is coupled to the supporting cradle using an elastic cowl mount assembly. The elastic cowl mount assembly includes a conical bushing coupled to the cowl component, an external housing coupled to the supporting cradle, and a compliant body coupled to the conical bushing and the external housing. The compliant body permits radial and axial movement of the conical bushing relative to the external housing.