"mattress," late 14c., paillet "bed or mattress of straw; small, simple bed," from Anglo-French paillete "straw, bundle of straw," Old French paillet "chaff, bundle of straw," from paille "straw" (12c.), from Latin palea "chaff," perhaps from PIE *pelh- "chaff," source also of Sanskrit palavah "chaff, husk," Old Church Slavonic plevy, Russian polova "chaff," Lithuanian pelūs "chaff."
"flat wooden blade" used as a tool by potters, etc., for shaping their wares, early 15c., from Old French palete, diminutive of pale "spade, shovel" (see palette, which is the more French spelling of the same word). The original sense in English was medical, "flat instrument for depressing the tongue." Meaning "large portable tray" used with a forklift for moving loads is from 1921.