It's worth noting that your source doesn't even have an entry for the programming sense of "parse", and so seems an odd way to back up your argument. AHD's entry for "parse" (4 ed., via dictionary.com) contains:
parse v. tr. ... 4. Computer Science: To analyze or separate (input, for example) into more easily processed components.
That seems to encompass what Anonymonk described as "parsing" -- analyzing (TCP/IP) input into more easily processed components. It's still not in my lexicon with that broad a sense, though; my take is probably closer to M-W sense 1.1, without restricting the domain to { things that have "component parts of speech" } (i.e. the first M-W sense without limiting it to "natural" languages).