Parsing is breaking down a clue into its parts and explaining the reasoning for the answer, various parts of the answer ('recurring = reverse the word', stuff like that). Some people are very good at it.
A lot of crossword solvers don't think the crossword can be considered to be "completed", unless they have solved the parsing!
Malone, incidentally , is very good at parsing, despite what she says!
Thanks for that, Elle. I am usually happy with my answers and know that they're correct, but I sometimes only parse them for someone else's benefit - and that's when I often miss out little bits!