The at89lp series is a bit different than avr or at89s with respect to isp. It requires an extra wire because it needs both a target reset and a target slave select signal (besides the common MISO, MOSI and SCK).
It is already a while ago but I once experimented with these chips, just out of curiosity. I built an ArduinoISP version that manged to program them. That version still sits in an old branch of my ArduinoISP git hub repo:
It can read and write flash of the devices I have: the at89lp4052 and the at89lp51ed2. It should work for your at89lp51rd2 too. Don't try tho change fuses, I dn't know what will happen (and it can brick your mcu). Instead use the companion at89lp_isp.ino sketch to set the fuses correctly for the at89lp51xxx.
If you care to pursue this, I recommend to start with the at89lp_isp.ino sketch. It allows you to burn a blink firmware to the at89lp51xxx as a first test. The led blinking 8051 code is built in, in the sketch.
For the at89lp4052 chips the factory fuses are fine but for the at89lp51xxx you need to set the fuses correctly in order for the code to start from address 0x0000. The at89lp_isp sketch can do that too.
You communicate with the at89lp_isp sketch by sending it serial commands: see code: 'p' enters or exits programming mose 'a' sets the fuses... You will first want to try whether you can read out the signature bytes.
Added 9 years, 1 month ago.