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Notice something obvious about a falling object: as it falls, it passes through the air. But all objects experience gravity; what's holding the air up? In other words, why doesn't air fall too?
The force that counteracts gravity and keeps the air up in the atmosphere is called the vertical pressure gradient force. (Technically, it's actually the vertical component of the pressure gradient force, fundamentally no different from the horizontal pressure gradient force which we'll get to later.) The air pressure is essentially zero at the top of the atmosphere and increases rapidly as you go down: every 3-4 miles, the air pressure doubles.
Imagine an air parcel again, floating in the air. The air pressure at the top of the parcel, pushing it down, is just a little bit weaker than the air pressure at the bottom, pushing it up.
This slight excess upward force is just enough to counter the force of gravity, that is, the weight of the air. So the air floats in equilibrium, just like the air around it.
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