**Abstract:**

We introduce two lambda-calculi and show that they are expressive for
two canonical domains of parallel functions. The first calculus is an
enrichment of the lazy, call-by-name lambda-calculus with
call-by-value abstractions and parallel composition, while in the
second the usual call-by-name abstractions are disallowed. The
corresponding domains are respectively Abramsky's domain of lifted
function space, and a lifted domain of strict functions. These domains
are lattices, and we show that the parallelism is adequately
represented by the join operator, while call-by-value abstractions
correspond to strict functions. The proofs of the results rely on a
completeness theorem for the logical presentation of the semantics.