If you don’t reclaim enough pallet lumber to justify a specialized prybar for the purpose, it is of course possible to knock down shipping pallets using everyday hand tools. A common mistake is trying to use a crowbar or the claw end of a hammer to pry the boards apart. This is almost certain to split and damage the wood, which may be OK if you just want to destroy the thing, but is self-defeating if you’re trying to reclaim the lumber.
Instead, pound out board and nails, from behind, using a hammer. Or better yet, a pair of hammers. This video from YouTuber miscpro demonstrates his practiced method for knocking a pallet apart using a common claw hammer, a small sledge, and a short scrap “spoil” board to take blows that would otherwise mar the wood being reclaimed. The video page on YouTube, linked below, includes some good written directions for the process, as well.