Direct parallel interfacing to an LCD is fairly easy for a PICAXE chip. That just requires a couple of routines which can output command and data bytes and strobe the correct signal lines.
Parallel via I2C to a PCF8574 is a case of working out the LCD signal mapping and modifying the PICAXE routines to suit that and I2C transfer.
Parallel via I2C to an MCP23008 requires the same as for the PCF8574 plus initialisation of MCP23008 registers and corresponding adjustment of the PICAXE routines.
Handling the MCP23008 is therefore more complicated but should be possible, though I do not recall seeing any proven code on the forum. The untested code and datasheet links posted by Westaust55 should be a help. Understanding how a PICAXE direct parallel LCD interface works and how and why the code needs to be adjusted for the PC8754 would be the best place to start and prerequisite for doing the same for the MCP23008.