My studies have suggested that παρουσία primarily does not have the import "coming" or "advent" when applied to Christ: it appears that παρουσία possibly could mean "arrival" in some contexts. In that connection, the word evidently refers to the visit of important personages in the Greek papyri. However, does παρουσία mean "arrival" in Matthew 24:3?
BDAG Greek-English Lexicon points out that παρουσία has these potential denotations: (1) "the state of being present at a place, presence" or (2) "arrival as the first stage in presence, coming, advent."
Sense (1) is clearly found at Phil 2:12 where Paul contrasts his own παρουσία with his ἀπουσία. 1 Cor 16:17; 2 Cor 10:10 also seem to be examples of παρουσία being used to mean "presence" although some believe that it may signify "arrival" in Paul's first letter to the Corinthians. See Rogers and Rogers, The New Linguistic and Exegetical Key to the Greek New Testament, p. 391. But compare Marion L. Soards, 1 Corinthians (New International Biblical Commentary), p. 364.
BDAG suggests that παρουσία in 2 Cor 7:6 and Phil 1:26 refers to the "coming" of Titus or Paul. However, one could just as well understand παρουσία in these verses as "presence" or "the state of being present at a place." See Moises Silva's work Philippians (The Wycliffe Exegetical Commentary), pp. 86-87. To witness examples of παρουσία employed as a terminus technicus (technical term) in the case of Christians and non-Christians, consult Moulton-Milligan, p. 497.
Regarding the use of παρουσία as terminus technicus for the "presence" of Jesus Christ, I find N.T. Wright's comments enlightening:
"But why should we think--except for reasons of ecclesiastical and scholarly tradition--that PAROUSIA means 'the second coming,' and/or the downward travel on a cloud of Jesus and/or the 'son of man'? PAROUSIA means 'presence' as opposed to APOUSIA, 'absence'; hence it denotes the 'arrival' of someone not at the moment present; and it is especially used in relation to the visit 'of a royal or official personage.' Until evidence for a different meaning is produced, this should be our starting-point" (Jesus and the Victory of God, page 341).
What does Wright mean by the "arrival" of Christ, however? In that same publication, he makes it clear that he is apparently referring to the "enthronement" of Christ and not to his so-called Second Advent:
"For the ordinary sense of 'arrival', cf. 1 Cor. 16:17; 2 Cor. 7:6, 7; 10:10; Phil. 1:26; 2:12. From this, the most natural meaning for the word as applied to Jesus would be something like 'arrival on the scene,' in the sense of enthronement" (ibid).
TDNT makes the point even clearer in its treatment of PAROUSIA. Finally, Louw-Nida Greek-English Lexicon shows that ERXOMAI possibly denotes: "to move from one place to another, either coming or going."