Yes and no. Chains and sprockets generally wear together. But as I said in the prior post, measuring stretch is only an indicator of chain wear, not a true measurement of that wear. Roller wear, which doesn't show up properly in the stretch measurement is just as important. In fact the best measurement of wear is how the chain actually meshes on the sprocket, but that's difficult to do reliably.

The best evidence that your sprockets are worn is that an older chain runs on them and a new one doesn't. You can confirm that with a new cassette of the same brand (don't toss the old stuff yet), and if your new chain runs on a new cassette of the same brand than you know for sure that your problem is sprocket wear.