Welcome to the forum.
Unfortunately not. The Pickit supports ICSP, this is a 5 wire interface (including ground) which is used to program microcontrollers with a serial data protocol. The W27C512 is a parallel interface device, which requires the full width of 8 bit data and all address pins controlled in order to program. The Sure ICSP socket board that you have there simply connects the ICSP signals to the relevant pins on different sized PIC microcontrollers.
You might try buying a TL866 programmer, but there are several variants of this and, not owning one, I don't know whether the W27C512 is supported and on which variants.