For anything covered in grease or oil there are proprietary cleaners, I have always used Gunk on greasy parts, and then wash off with water. Petrol is dangerous to the skin as well as the fire risk, not too sure about paraffin.
What ever you use, get some of those rubber gloves that paramedics and Kwik fit fitters use.
I am in the process of doing the same thing. Lewmar recommend using parafin or kerosene, I will be using stuff called Lowtox (because I have easy access to it) which is white spirit based I believe.
Kerosene is almost completely safe and does the job no problem. For winch parts I use it in conjunction with a cheap paintbrush with the bristles cut short so that they are stiff.
For myself, I would not use petrol, partly because of the claimed additives vs skin matter but mostly because of its flammability (same with white spirit which in some countries is hard to buy now because of its flamability) and kerosene being a perfectly good alternative.
Diesel works too, but is messier and smellier to use.
Careful how you take it/them apart, dont lose any bits.(over the side) and dont over grease the bits when you put it/them back together.
Winch, remines me, my mast winch needs doing.