In two steps:
Betz limit 16/27 is 4 times the limit in pumping mode which is given for 4/27. The paper details the reasons. I see a simple reason: the swept area, in pumping mode, is dragging at 1/3 wind speed which is reel-out speed, keeping 2/3 wind speed. So the force (measured by the square of wind speed) becomes 2²/3² = 4/9. Power is force x reel-out speed: 4/9 x 1/3 = 4/27.
The parachute takes the whole area of power available in the wind, if we do not take into account wind deflections around the parachute, which remains to be seen (but it can be the same for the swept area of a conventional wind turbine). Now if we increase the drag coefficient Cd of the parachute, the drag force will increase. And said drag force will affect the swept area as well as the flying body (in this case the parachute), both being merging. If the Cd is 3, the AWE parachute system could harvest up to 12/27 of the power available in the wind.