This article extends tail-call optimisation by applying it to asynchronous calls. We first introduce Tail-Modulo-Await, a novel code transformation for asynchronous tail recursive functions that prevents the creation of unnecessary tasks. We then show how to combine Tail-Modulo-Await with the existing Tail-Modulo-Cons optimisation; we obtain an optimisation able to turn a recursive function with multiple tail calls under constructors into a parallel version of the function, also optimised in space. We formalise both optimisations over representative calculi, and prove them correct through backward simulations. Finally, we provide a proof-of-concept implementation as an OCaml syntax extension and evaluate it experimentally, showing our approach optimises both memory and execution time.