Definitely don't put ponies in direct contact with stained wood!
Even years later it can still transfer dye. Vinyl is super duper happy to pick up dyes of all kinds. I wouldn't suggest just parchment or wax paper - they're porous, and while better than nothing can potentially act as a transfer medium themselves over time. Get something completely water/stain proof, like Mylar (the archival gold standard) or other stable plastics, avoid PVC.