Actually, it's Ballistol, not Barristol. Important if you do a websearch to find it.
Pedantry out of the way, Ballistol is basically harmless to wood and synthetics. It is even touted as useful in wound healing. Like CLP, it's meant to be both a cleaner and lubricant. And like CLP, it is less than ideal in either role. I found Ballistol to be a pretty decent cleaner, especially for blackpowder and corrosive-primered ammo, but I didn't think much of it as a lubricant. My opinion of CLP is opposite: good lube, not so good cleaner.
Practically speaking, everything petroleum-based is potentially harmful to some wood finishes and some plastics. However, most modern finishes and plastics are resistant to all but the most aggressive solvents.
Over 2 decades of liberal CLP use has not harmed the finish on any of my firearms, including wood-stocked milsurps finished with tung oil or linseed oil.
When it comes to dedicated cleaners (aka solvents), the same cannot be said. Ed's Red (for example), is a very good and dirt-cheap solvent/cleaner, but it will attack stock finish. Just wipe down everything when you're done cleaning and you'll be fine.