It is an adiabatic process because the rising parcel of air is large and therefore has a small surface to volume ratio.
As the parcel of air rises the pressure exerted on it by the surrounding air gets less and less. Therefore the parcel of air expands and cools adiabatically by pushing out against this atmospheric pressure. The energy it loses in this way is proportional to the volume of the air parcel.
Of course it also loses heat by conduction but the rate of heat energy lost by conduction is proportional to the surface area of the parcel of air.
The ratio of heat loss by conduction to heat loss by adiabatic cooling, equals the ratio of surface area to volume, which is very small for the large parcels of air which form clouds.