1540s, "words, clauses, etc. inserted into a sentence, not grammatically connected to it but explaining or qualifying a word," from French parenthèse (15c.) or directly from Medieval Latin parenthesis "addition of a letter to a syllable in a word," from Greek parenthesis, literally "a putting in beside," from parentithenai "put in beside," from para- "beside" (see para- (1)) + en- "in" + tithenai "to put, to place" (from reduplicated form of PIE root *dhe- "to set, put").
By 1715 the sense was extended from the inserted words to the two upright curved brackets (parentheses) used by printers or writers to indicate the words inserted.
Your first figure of tollerable disorder is [Parenthesis] or by an English name the [Insertour] and is when ye will seeme for larger information or some other purpose, to peece or graffe in the middest of your tale an vnnecessary parcell of speach, which neuerthelesse may be thence without any detriment to the rest. [George Puttenham, "The Arte of English Poesie," 1589]
A wooden parenthesis; the pillory. An iron parenthesis; a prison. ["Dictionary of Buckish Slang, University Wit and Pickpocket Eloquence," London, 1811]