Problem is, the 80's definitions of parallel (more than one bit at a time) and serial (one bit at a time) really don't apply any longer. PCI Express, for instance, is clearly a serial bus, even tho it can run 32 bits in each direction at once.
A more modern definition of serial, I'd argue, is a bus where there are no hardware-defined command, address, or data signals - just bits getting passed, which could be any one of those three things depending on context and regardless of which wires they're on. Parallel, by contrast, is a bus where a particular wire is pre-determined in hardware to send command or address or data signals.
By my rule, then, PCI Express is clearly serial, as are SATA, USB, and Ethernet. The DDR memory interface, PCI, PCI-X, and AGP are all parallel interfaces.
Fencer's got the right idea then, about why serial's taking over the world. Parallel busses are limited in their switching speeds by the need to maintain timing relationships between the various signals (the command signals, for instance, need to do their thing when a particular unit of data is on the data lines). Serial doesn't have that restriction; it can just switch the bujeezus out of the signals (2.5Gbps for first generation PCI Express, and it'll go up to the signaling limits of copper), and let the state machines at each end figure out the context of each bit and what to do with it.
Those state machines are NOT simple - they take a lot of transistors, which is part of why we haven't been high-speed serial all along. Now we're getting the transistor budgets to do that sorta thing, which is why serial's now taking over the landscape.