Does having multiple quantum computers in parallel speed up Shor's Algorithm? Essentially, how useful is it to have multiple quantum computers in parallel if the goal is to, for example, recover a 2048 bit RSA key?
 A: Yes, parallel computing can reduce the time it takes to perform Shor's algorithm.
For example, the meat of Shor's algorithm is a modular exponentiation. Which you could perform via N conditional multiplications one after another. Instead of that, you could do $\sqrt{N}$ groups of $\sqrt{N}$ conditional multiplications, then do $\sqrt{N}$ multiplications to combine the intermediate results together into the final product, then uncompute the intermediate states with $\sqrt{N}$ groups of $\sqrt{N}$ conditional un-multiplications. This reduces the multiplication-depth from $N$ to $3 \sqrt{N}$.
Of course that's just a simple example. There's lots of other stuff you can do. Pick your favorite classical parallel binary multiplier circuit, and derive a quantum circuit from it. Or do even more multiplications in parallel.
That being said, quantum computers already need to operate every qubit constantly in order to perform error correction. So they are going to naturally be operated in a very parallel way. There's no need to wait for a second or third computer before you start thinking about parallelization.
