Parentheticals, it is fair to say, are everywhere, and, as can easily be shown, they come in a huge variety of flavors. Two characteristics set them apart from ordinary arguments and adjuncts: they tend to show some freedom of position within their hosts, and they tend to be semantically opaque to scope of quantification, negation, and long-distance dependencies. This study demonstrates that parentheticals are independent pieces of language that are jammed into their host, subject to prosodic constraints, and they stand in some discourse relation to the host, serving as a side comment or as supporting material. But, crucially, parentheticals are "asyntactic": they are not syntactic constituents of their host. The analysis is couched in terms of Jackendoff's Parallel Architecture, in which phonology, syntax, and semantics are independent but linked generative systems, and in which words and rules of grammar are encoded in a common format. In support of the asyntactic treatment of parentheticals, this study develops accounts of prosody and discourse structure, components of the grammar that are new to the Parallel Architecture. The work presented here is innovative in several respects. First, it addresses and formalizes a wide range of parenthetical types, where most previous studies have dealt with only a limited selection. Second, it integrates the prosodic, syntactic, semantic, and discourse aspects of parentheticals, where most previous studies have dealt with only the prosody or only the semantics/discourse structure. Finally, many aspects of the analysis give reason to prefer the Parallel Architecture over other theories of grammatical structure.