My referent for the parts was the bonus track on The Turn of a Friendly Card that's just Chris Rainbow's vocal parts. Comparing that with the final version reveals that I have a lot less than I thought I did. I have up until about 0:40 on that track (and the first ten seconds or so is just Rainbow's getting ready), but I think that's only half of the backing vocals for the verses in the final version.
Anyway, in that first forty seconds, there are four parts, and each is four measures long. Every four measures, a new part is added. After adding all of the parts, I sang them a fifth time too.