If an unsaturated air parcel ascends adiabatically in the atmosphere with an initial temperature, would the final temperature of the air parcel, when it reaches it's destination, be the same at the initial temperature since it does not exchange heat with it's surroundings?
No. The parcel of gas expands adiabatically as it rises. This makes it cool.
Here's one way of looking at what happens: No heat is transferred to or from the parcel of gas (so, adiabatic). At the same time, the parcel of gas expands as it rises. That means the parcel is doing work on the external environment. The temperature of the parcel must drop to conserve energy.