What kind of glue can I use in my bathroom if I'm using pallet wood?

Liquid Nails plus nail into studs.

Locktite Power Grab. Also nail into studs.

I agree that nailing your pallet wood into the studs in your wall is a great idea. But when there is no stud available you might need to use an adhesive. I would use either Liquid Nails or Gorilla Glue.

If either the walls or the pallet pieces are not straight, no amount of glue is ever going to hold them all in place without cracking, slipping down, or popping off. You will need nails for sure. Where there are no studs, use toggle bolt screws.

Here's a collection of projects adding pallet wood to the walls. Maybe that will give you some helpful hints.

https://www.hometalk.com/search/posts?filter=pallet%20wall

If you ever want to remove the paneling, the glue is going to destroy the walls. Use nails instead. Get an inexpensive nail gun to attach the wood. It's fast and fun to use.

I’m with Emy, try to nail rather than glue as glue will destroy a wall if you or someone else ever wants to remove it. You can rent the tools if necessary. I attached lath to a bathroom wall that way, you can see it here: https://www.flippingtheflip.com/2017/11/hall-bathroom-revisited.html?m=1