皮 is the indexing radical and gives the literal stage for the action: skin or surface. Wrinkles and creases are by definition things that happen to a 皮 - whether human face, fabric, or paper - which is why 皱 needs this body-surface anchor.
刍 supplies the sound, drifted from chú to zhòu. The same phonetic core feeds 雏 (chú, chick) and 趋 (qū, hurry). Here the cut-grass imagery also adds a faint hint of bunched, gathered material - fitting for wrinkles and folds.
