If you really want to use ds80c320, and EEPROM, even after being told there are devices with inbuilt flash and Boot loaders for a couple of dollars, then these are the issues you will face.

What you ask can be done
a) ds80c320 has no ROM, so it always needs to be fed code externally.
b) EEPROM cannot supply opcodes whilst it is programming

Thus the solution is to use TWO EEPROMS, (these can be stacked, with a couple of pins separated) and use some means to select.
One chip (use read only or ROM) holds the Boot loader code, you first load this using your parallel programmer, with Boot code you find/modify.

The other chip is for your changing code.

You add a simple Mux (eg google 4 x 2:1 Multiplexer) to flip between Boot and run, via a switch, or using PC-uart handshake lines.

During PGM phase, DS80C320 fetches opcodes from your ROM EEPROM, and uses /RN and /WR to access the Other EEPROM (RUN).

Then for run, your MUX swaps RUN.OEN from /RD to PSEN, and the other ROM.OEN can disable, or swap conversely from PSEN to RD if you want to include tables, or fonts, or similar in the ROM chip.
A reset now starts from RUN.

ROM.WR is disabled, (pgm that in your programmer) and RUN.WR is either always connected (dangerous), or connected to uC.WRN only in Boor, and disabled in Run.