The 12F629 is programmable on the Pk2 via Mplab or V2.61
The 16F54 is Not programmable by the Pk2 at all, you need a Pk1 or Pk3 for that; its such an old chip you would be better to bin it than struggle on.
You words so far have confused us, so please give us some hard evidence.
First , when you use the Pk2, what program are you running it with, MplabIDE version ? MplabX, Pk2 V2.61 or something else ?
If you connect the 12F629 to your Pk2 and READ the 629 what happens; if it gives an error please post a screen shot of the messages.
If it reads ok, and you try and program the chip, what is the actual error message, again a screenshot help.
Importantly , Please provide a drawing or picture of how you have connected the Pk2 to the 12F629
A picture / brand and model name of your Pk2 clone would possibly help as someone might recognise it and know for certain that it is or is not capable of HVP.
It would be so unusual for anyone to make and sell a pk2 clone without hvp.