Parsing used to be done on sentences by schoolchildren (do they still have to draw those diagrams on the chalkboard?), to show that they understood which words were nouns, verbs, and so on, and what part each word played in the sentence. Used in programming, it means much the same thing.

From dictionary.com:

4. Computer Science. To analyze or separate (input, for example) into more easily processed components.

So if you're talking about parsing an HTML file, you'd probably be picking the file apart and somehow identifying which pieces are what HTML tags, which pieces are normal text, etc.