Parsing, in its grammatical sense, was an important part of studying languages in the past. It is rarely called that these days, and the exercise is rarely used now in British education. It involves looking at a sentence and identifying the word class to which each word belongs. ('To parse' is derived from the same root as 'part'.) In addition, students note the rules that govern the word and its inflections. This is a more important element of parsing in highly inflected languages, especially the classical languages Greek and Latin, where the exercise was most practised.
You are more likely nowadays to see the word in terms of computer languages and programs. OED gives its meaning as "To analyse (a string) into syntactic components, esp. to test conformability to a given grammar." it is beyond AWE's current scope to say more.