Rain that is running off something into your pool, depending on what it is running off of, can adversely affect your pool's chemistry, clarity and sanitation, depending on what the runoff is dragging into the pool with it. For an above ground pool, that usually doesn't occur. Maybe some water off a surrounding deck, but that's usually not a big deal, and would only occur if the deck wasn't built to drain away from the pool, as it should.
Rain water that just falls into a pool doesn't do much, unless it is so much rain that it causes the pool to overflow. If rain just falls in a pool, it can affect pH a little, but not much else. The added water will eventually evaporate, and the pool will be back where it started, chemically speaking. If the rain's pH is different than the pool's, then, yes, the pH might be affected, but not my much and it would be easily corrected with some dosing of simple chemicals.
If there is so much rain that the pool overflows, then the additional rain water will dilute the pool water, and all the chemical balances would likely be affected to some degree. So you'd test all the levels, and adjust each as needed. If you use the TFP method of testing and dosing, "fixing" a pool after a rain is a simple thing.
Now if we're talking about a big storm, that blows leaves and other debris into the pool, that's another matter. (But still correctable with the right know-how and some effort.)