Parsing in C++ is more fragile (especially with the insane algorithms we're using to perform this parsing, for the implementation-defined parts of it), and we have to worry about things like bug 866608. Something to do after bug 1265136 lands, since that's (probably) more important than doing this. Once this is done, we probably can self-host Date.parse and the Date constructor as well. Given just self-hosting this imposes costs on those algorithms, it may be that we should do all of them in one bug (in separate patches for readability, and probably in separate revs landed all at once). The C++->JS overhead may not matter for these, but it doesn't seem worth running the risk.
Tentatively taking, as something that's easy to work on while a laptop's being repaired -- might dump back in the pool with partial progress when the new one arrives (given that I have other work to do, that would be more pressing, if it were comfortably doable).