What kinds of parallel RAM are there?
The term you want is "dual port RAM", and yes, they exist both as discrete components and as building blocks inside chips such as FPGAs. They are frequently used when it is necessary to pass data from one clock domain to another, and somewhat less frequently as cache storage, which is what you seem to be trying to do.

DDR/QDR (double data rate, and quad data rate, respectively) are something else: they are techniques for improving the bandwidth of the individual connections to a RAM chip.

You can effectively create a memory with two read ports by simply hooking together two normal single-port memories. Connect their write ports to the same source, and use their read ports independently. The contents will always be identical copies.