To parse means to break things into parts. To parse a sentence means to break it into the parts of speech, so that you can see the structure of the sentence. There are eight parts of speech in English:

The Functions of the Eight Parts of Speech in English
Part of speech Function
Nouns Represent a person, place, thing, or idea
Pronouns Stand in for a noun
Verbs Represent an action or state of being
Adjectives Describe nouns
Adverbs Describe verbs, adjectives, clauses, phrases, or whole sentences
Prepositions Link a noun phrase to turn it in to something that acts as an adjective or adverb
Conjunctions Link words, phrases, clauses, or sentences
Interjections Express emotion

To parse a sentence means to break the sentence down into its parts of speech. The word parse comes from the Latin word pars, which means part.

Once you learn to parse, it becomes much easier to analyze the structure of a sentence. Once you can do that, it becomes much easier to understand the meaning of complicated sentences. When bad writers learn to parse, they suddenly find it much easier to turn confusing or ugly sentences into clear and beautiful sentences. Their writing improves, and even their thought processes become clearer.
Here’s a passage from the opening of Lewis Carroll’s Alice’s Adventures in Wonderland. All of the nouns have been underlined.

Alice was beginning to get very tired of sitting by her sister on the bank, and of having nothing to do: once or twice she had peeped into the book her sister was reading, but it had no pictures or conversations in it, ‘and what is the use of a book,’ thought Alice ‘without pictures or conversation?’

So she was considering in her own mind (as well as she could, for the hot day made her feel very sleepy and stupid), whether the pleasure of making a daisy-chain would be worth the trouble of getting up and picking the daisies, when suddenly a White Rabbit with pink eyes ran close by her.