To parse is to analyze something in an orderly way. In linguistics, to parse is to divide words and phrases into different parts in order to understand relationships and meaning. For example, English students are sometimes asked to parse a sentence by dividing it into subject and predicate, and then into dependent phrases, modifiers, and so forth.
In general, to parse someone's writing or speech simply means to interpret it.
In computers, to parse is to divide a computer language statement into parts that can be made useful for the computer. A parser in a program compiler is a program that takes each program statement that a developer has written and divides it into parts (for example, the main command, options, target objects, their attributes, and so forth) that can then be used for developing further actions or for creating the instructions that form an executable program.