These guys are absolutely right. To take the example further, lets say you have this math expression:
when you parse this, it becomes
So parsing something really is about breaking it down into its individual chunks and pieces. For instance, a parser might break your c++ program into functions, or your word into letters. If you have a whole made of parts, the parser looks at those parts. Make sense? Check out the definitions linked above, they're probably better at explaining it than I am. Hope that helped a little.
PS The above is a "parse tree", certainly not the only way to parse or represent a parse. Just so you know.