Parallel yes, as other people mentioned, this is almost the entire point of using FPGAs. Regarding Asynchronous, it depends on what you mean. Xilinx(AMD)/Altera(Intel) FPGAs are designed from bottom up to be synchronously clocked. The fabric and tools are designed to use synchronous pipeline registers everywhere to minimize combinatorial logic and increase throughput. You might want to have a design with multiple asy…
Just the regular advantages, only with FPGA, which means one can choose how logical elements are interconnected, and what's the logic of the chip. Among regular advantages are absence of clocks (less devices, no need to synchronize...) and energy is used when and where the switching happens.
A friend of mine unsuccessfully tried to squeeze asynchronous designs into some mainstream FPGA a few years ago. Tooling wasn't cooperative, and when he used some workarounds to avoid generating clocks, it was simply crashing. I don't think it's useless or for lack of trying - but asynchronous circuits in FPGAs are certainly not common.