Why are most parallel interfaces half-duplex and serial full-duplex?
For a true full-duplex interface, you really need two wires for each signal - one in each direction. This quickly becomes impractical for a parallel bus.

If you have a parallel bus, the bandwidth is often important. If you must add twice the number of wires, you can almost always benefit more by sending twice the amount of data instead of reserving half the wires for one direction and half for the other.