Discourse markers exhibit a range of grammatical properties that set them apart from many other lexical and grammatical forms. A number of hypotheses have been proposed to account for these properties. Most commonly, such accounts have drawn on grammaticalization theory, less commonly also on the notion of pragmaticalization. As argued in this paper, however, the rise of discourse markers cannot be described exhaustively in terms of parameters of grammaticalization or pragmaticalization. Looking at the rise of two English and one French discourse marker, the paper argues that this development is also shaped by a general strategy of discourse processing called cooptation, which accounts for properties of discourse markers that are beyond the scope of grammaticalization theory. Thus, discourse markers are described best as grammaticalized (paren)theticals.