The phrase “parts cross” means that between two parts, the higher one moves downward in pitch so far that it becomes the lower one (or the lower part moves above the higher part).

For example, in this passage for two clarinets, clarinet 1 begins on a higher pitched note than clarinet 2. But by the end of the phrase, the clarinet 1 is playing at a lower pitch than the second clarinet (see the boxed area).  

crossing parts in a score