JAKARTA, Indonesia—AirAsia didn’t have permission to fly from Surabaya to Singapore on the Sunday morning that Flight 8501 crashed into the Java Sea, Indonesian officials said Saturday.

The development came as Indonesia’s search-and-rescue agency said it had discovered four large pieces of the plane on the floor of the Java Sea. Because of that discovery, the search area was narrowed on Sunday and divers were able to enter the water to look at what was found, an Indonesian search-and-rescue official said. He added that weather...