When an air parcel rises and cools adiabatically, it is said that there is no heat transfer as work is done on the surrounding atmosphere as the parcel expands. The parcel loses internal energy and condensation occurs. I do understand this concept, but why is it that work is done on the surrounding lower temperature particles by the higher temperature particles and not simply conduction, which would be a heat transfer? Wouldn't the faster moving particles in the air parcel not conduct heat on collision?
The work being done on the surroundings is because the air parcel expands as the pressure decreases.
There would be some heat transfer if the parcel of interest has a different temperature than the surrounding air. This effect is smaller because gases are poor conductors of heat, as mentioned by @gerrit. We take advantage of this with fiberglass and styrofoam insulation.
I know that this is an old question and another answer was provided but I would like to elaborate on it deeper as I also find it to be an interesting question.
I think there were actually two questions that were formulated.
Why does the rising warm air mass not transfer its heat to surrounding cool air directly? I think the author meant two mechanisms here: conduction and diffusion.
Conduction. Direct conduction of heat from the warm gas to the cool gas is not an efficient mechanism since:
The gas molecules can only transfer thermal energy via random collisions. Therefore, each molecule has to wait until it collides with something before it has a chance to transfer its energy. This is unlike solids with, say, a rigid lattice structure where each atom's movement quickly impacts all nearby atoms in the rigid lattice.
In a mixed gas like air when two molecules do collide, the collision may not transfer all energy. When a light gas molecule collides with a heavy gas molecule (or vice versa) both particles may leave the collision with more or less the same kinetic energy they came in with (depending on mismatch in their respective masses). This is similar to impedance mismatch.
Diffusion. Direct diffusion of the warm gas molecules into the cool gas molecules is also rather slow especially compared to the dimensions of air masses affecting the weather. Diffusing molecules follow Brownian motion which expands only as square root of time. 
Why is any work done by the rising warm air?
- We concluded that the warm air mass is not diffusing into cool air and is not transferring heat to it directly. So what does it do instead? The warm air is pushing away the cool air like it would push a piston so it is doing work and hence loses internal energy.