Staining an Old Piano?

Sorry to tell you, but yes, it must be stripped to stain.

Before stripping paint check whether the finish is veneered as may not be worth stripping unless you reveneer.Many old pianos and organs used cheaper woods underneath various timbers different colors so veneered in walnut etc. depends on how far you wish to go on your restoration.You may be lucky that they have used solid timber would tend to be an exception rather than the rule i suspect.

Stain works differently to paint, it soaks into the wood and colours it so you can still see the actual grain. So unfortunately you'll have to strip all the paint off to stain it! Paint covers whatever's there before, you never know what you might find underneath.