I'm the paper guru!
Since I work in a Paper Mill I will answer this one! Wrapping parts in paper is not going to hurt your parts. As stated before, putting masking tape-which is paper and adhesive glue, will dry out and become very difficult to remove. You have to use the right kind of paper. Do not use newsprint. Do not use any paper that has printing on it. The ink in the paper will transmit over to the other surface and mar your new finished painted products. Avoid wrapping anything with a colored paper as stated before. There are several different kind of dyes that we use in papermaking some are acidic and some are alkaline based. These dyes could also mark your painted paper. I would use a packaging paper as they usually are made with little to no chemicals in them due to cost reduction and they are not dyed either. They would appear either just straight brown paper or a slightly bleached verison. White (bleached) paper can be used but would be more expensive. Never wrap steel parts with paper. I used to make a special paper called VCI (Vapor Corrosion Inhibiture) made specially for metal parts wrapping in bearing packaging and steel parts packaging. It is neutral pH-meaning not having any acid in the sheet, had no sizing-water repellancy, and has no other chemicals in it. It would then be coated with a corrosion proof chemical so when it was wrapped, the parts would not rust. Hopes this helps you out.