Well in this case the engine model/spec is not going to help as it is not an engine part. Need the mower model number at least. With that said the pulley may not be available separately from the transaxle; won't know until we can look up the mower.
That transmission is not for the Toro model 20371 you posted. You have a single piece pulley and the model you posted has a two piece pulley. 21038 is not a good Toro model number. Something does not make sense.
I bought this mower about two months ago from a neighbor woman I didn't know, she said it ran, it didn't but for $25.00 I t was still worth the money to me.
I replaced the Carb & gaskets, changed the fuel line and cleaned the gas tank and mowed the lawn for two months. When I was mowing another lawn on a hill I had to help it up the hill.
When I returned home I checked the belt that is about 1/4" wide and slipping. I am waiting for a new belt. While working on it I found a chinese gasket and a bread tie for a bolt and fishing line connecting the drive cable to the handle. someone may have changed the transmission assembly.
Another question, I was watching a YT video, the mower in the video had a Belt cover under the Blade. Is the mower I have missing that part or did Toro not include it on some models?