Regarding the clip used on The Last Leg - I think it's a grey area and it's all down to context. The clip used contained a funny moment from parliament which was being shown and presented as-is, I think this is ok with the rules. If it was a serious clip that they were making fun of, then that would be against the rules. Making fun = BAD, Containing fun = GOOD...
The clip which kicked it off on Last Week Tonight was a montage of clips of the speaker calling MPs out in funny/very British ways - so probably bad.
Looking at the actual rules (link below), no light-entertainment or political satire programmes can use footage (6a) but programmes CAN use footage if they are "magazine" type shows that also have "music or humorous features" as long as "the different types of item are kept separate" (6b).
Last Week Tonight and HIGNFY would be stopped by rule 6a but The Last Leg would be able to go ahead provided they respected rule 6b.