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Ask yourself this
Are they attaching anything new into the party wall?
Are they making holes in the party wall and fixing structural stuff via the party wall?
Are they adding additional load to the party wall?
Then yes, probably.
If they are making a repair / remedial to an existing party wall and keeping it within its exiting design and materials, then no.
(didn't initially grok that it was your neighbour who were doing the works - you should talk to them to better understand their plans so you can answer the questions ^ )
It depends what they are doing as to whether they need PWA. As for the parapet wall they could say that it fell down as they were working near it and are simply rebuilding it and trying not to worry you. If of course could be a lie.
But I certainly wouldn’t be letting them work off my roof.