to refuse to listen, and so to disregard or disobey
The word is built from the ordinary verb "to hear" plus a prefix meaning "beside" or "past," so at root it pictures hearing a word off to the side rather than straight on. From that single image the meaning runs in two directions: hearing something by accident (overhearing or mishearing it) and hearing something and then turning away from it (ignoring it, refusing to heed it, disobeying). The base verb it is built on is ἀκούω, plain hearing.
By the time of the New Testament the second direction had almost entirely won out: the word normally means to refuse to listen, and therefore to disobey. This sits on top of a Hebrew habit of speech in which "to hear" already carries the sense "to obey," so that "not hearing" is disobedience. That is what gives the word its weight. To refuse to hear an authoritative word is to rebel against it. It is the exact mirror image of the word for obedient, submissive hearing (ὑπακούω).
The clear New Testament uses follow this pattern. In Matthew 18:17 a person who "will not listen" even to the gathered community is to be treated as an outsider; the refusal is willful and settled. The one genuinely debated occurrence is Mark 5:36, where Jesus, as the report of the girl's death is spoken, either disregards it or overhears it, or possibly both at once. The related noun (παρακοή) carries the settled sense "disobedience."
Nuances
What παρακούσας means in Mark 5:36: whether Jesus disregarded the report of the girl's death, overheard it, pretended not to hear it, or whether Mark chose the word so that more than one of these is meant.
- majorityJesus disregarded the report, sovereignly refusing its verdict that further help was pointless; the majority modern reading, resting on the Septuagint and wider Koine pattern and on the countermanding force of his immediate command not to fear but only believe.; Robert A. Guelich, Mark 1-8:26, WBC 34A (Dallas: Word, 1989), 302-303; Adela Yarbro Collins, Mark: A Commentary, Hermeneia (Minneapolis: Fortress, 2007), 284-285; R. T. France, The Gospel of Mark, NIGTC (Grand Rapids: Eerdmans, 2002), 239-240; J. P. Louw and E. A. Nida, Greek-English Lexicon of the New Testament Based on Semantic Domains, 2nd ed. (New York: UBS, 1989), 36.26
- minorityJesus overheard the word spoken to Jairus; the older perceptual sense, supported by the direct-object construction and by the fact that the message was addressed to Jairus rather than to Jesus.; Joel Marcus, Mark 1-8, Anchor Bible 27 (New York: Doubleday, 2000), 366-367, 370; Liddell, Scott, and Jones, A Greek-English Lexicon, 9th ed. (Oxford: Clarendon, 1996), s.v. παρακούω
- contestedThe word is ambiguous by design, so that both overhearing and disregard resonate together.; Morna D. Hooker, The Gospel According to Saint Mark, BNTC (London: A&C Black, 1991), 149-150; Ezra P. Gould, A Critical and Exegetical Commentary on the Gospel According to St. Mark, ICC (Edinburgh: T&T Clark, 1896), 100