When Dr. Herbert Pardes retired as president and chief executive of NewYork-Presbyterian Hospital in 2011, the institution honored him at its annual "Cabaret" fund-raiser. More than 1,000 guests dined on wild mushroom soup catered by the restaurateur Danny Meyer and listened to Kelli O'Hara, a star of "South Pacific," serenade them with Rodgers and Hammerstein, Sondheim and Berlin. But there were more thanks to come. Three years after retirement, Dr. Pardes is still employed by the hospital as the executive vice chairman of its board of trustees, a position that compensation experts say is rare in the nonprofit world, though much more common in for-profit companies.