are there any software that can do the ====> part.
Dunno - that would pretty much depend on the client
in your diagram
- and the local http server
As stated before, you aren't going to find software that can perform magic and parallelize stuff without support from it's endpoints... and you provide very little information on goals and involved software, so it's hard to give any concrete recommendations.
So, what's the deal? Is this understanding correct?
1) high-speed server preFETCHes HTTP resources
2) server PUSHes those resources to "client" machine
3) "client" machine runs a http proxy, and web browser on "client" machine access that local proxy
...and you want the PUSH to be faster than it currently is, and you don't want to wait for entire FETCH to be done before you PUSH?