First usage of parentheses or brackets ( and )

Sorry, but according a (1993) book (Pause and effect: an introduction to the history of punctuation in the West (by Malcolm Beckwith Parkes)) Erasmus didn't invent parentheses (but he gave them a nice name).

A form of parenthesis had already been used in (a manuscript called) De nobilitate legum et medicinaue (from 1399) that looked like a Γ and >.

In a (1428) copy of (Cicero's) Epistolae ad familares, the marks were paired off as <>.

They subsequently became the familiar round brackets (()) (as recommended by (Gaparino) Barzizza (1359-1431) (in (his) Doctrina punctandi)).

Edit: Furthermore, if you re-read Wikipedia you'll see it says (citing (Lynne) Truss):

Desiderius Erasmus coined the term lunula to refer to the rounded parentheses (), recalling the round shape of the moon.

He coined the term lunula. It makes no claim of (Erasmus (or anyone else) inventing the mark.

(Further to Hugo's answer,) John Lennard (a linguistic scholar) talks about these in his book, The poetry handbook and (in a footnote) says that Erasmus coined the term in the year 1531. My understanding (from the Wikipedia article on him) is that by this time he was indeed living in Basel - but (as Hugo notes) here he coined the term lunula but did not first use brackets.

They go back as far as the inverted "Nun" letters bracketing certain passages in the TaNa"Ch (Hebrew Scripture): Genesis 6-; Psalms 107; Numbers 10:35-36. Later scholarly readers of these ancient texts in their original ported this into their European languages, in shapes like <> and () that worked with Latin-derived letters' form.

Just to add: Salutati visually translated the classical figure of digression "parenthesis" or "interpositio" by inventing the typographical mark which would bear the same name. So the stylistic device of inserting additional stuff into the main thing has been around since Cicero and Quintilian.

In the Renaissance, parenthesis referred to three things: the literary device, as well as the words within the typographical marks (parentheses or round brackets).

Also, just as a quite interesting aside, Salutati added the early parentheses by hand into the manuscript copied or noted down by his assistant. It was so important for him to control the flow of the sentence that he actually made the effort to read the text and sequeeze the marks in himself. Punctuation was considered that powerful that the man himself needed to do it, or was allowed to do it, not his secretary!

P.P.S. Salutati was a punctuation fan and also introduces the first exclamation mark into the 1399 manuscript (though he didn't invent it).