If you are talking about the un-coated butcher paper it will not work. Coated freezer is the best and can be found in most supermarkets. Parchment paper or wax paper will both stick to soap. If you are hp'ing you may get away with uncoated
Peach paper is also known as Pink Butcher Paper, and it is designed for use in smokers and with heat, but it does not have a poly-coating that makes freezer paper 'leak-proof'. See this link. If I had any, I'd try it and see how it works, just to verify that it would leak. I suspect it is fine with HP soap that isn't super fluid.
I did use parchment paper a couple of times, and warn against using the kind with a design. The dye from the design transferred to the soap and I was not happy as it was not pretty; mostly it looked like it had dirty spots on the surface of the soap.
With peach paper, I'd be less worried about leakage and more concerned that the paper will not peel cleanly away from the finished soap. The same trouble will happen if you use freezer paper with the dull side facing toward the soap or waxed paper.