I've been using this site to get part numbers due to their extensive parts diagrams as I convert a P216012 to the Honda GXV160, so I am going between the P216012 and a CP215520HV (7800849-00).
You are working on much newer machines than I, a series 12 and 20 I think from your numbers. Yup that web site is good.
On my series 4, they list or show most every part but the throttle cable part number. I went up to a series 6 or 7 I think it was to find a listed part number. Cool thing about Snapper, I think, all the series 0-20 mowers share the same parts for the most part. Great design, no need to change anything.
slomo