I have a PIC16F (Microchip) SOIC chip
which PIC16F, exactly? Older 16F chips used a high voltage programming algorithm that would probably be annoying to set up (about as bad as HVSP for a tiny85), but the newer chips mostly come with low voltage serial programming enabled. which OUGHT to be pretty easy. Whether you can get a tool comparable to avrdude to TALK to the programmer is a separate problem. There's a long legacy of PIC programmers (and software tools) that were pretty "weird", and I don't know that there has been as much standardization and documentation of communications protocols as there has been in the AVR world.
Probably better to spend $20 for a "Microchip SNAP" programer/debugger (it won't do HV either, though.)