I don’t think Triumph ever had an oval engine cover, although that doens’t look exactly like the BSA ones, maybe it’s one of the obscure makes, I’m pretty sure it’s not Norton, Ariel, Matchless or Enfield.

It’s similar to the AJS Model 16, but that had a single exhaust, this has two.

I think it’s a lightning, until I visit my parents again and check their classic bike books.

static.howstuffworks.com/gif/1970-bsa-lightning-1.jpg

That’s a very god shout, and would be suitably obscure, but I can’t find the model, and the engine case still doesn’t look right. BSA is flat heads, the Bantam was one, but I’ve gone off the idea of them anyhow, there’s something odd about the rear swing arm.