early 15c., "alleviate (a disease or its symptoms) without curing," from Medieval Latin palliatus, literally "cloaked," from past participle of Late Latin palliare "cover with a cloak, conceal," from Latin pallium "cloak" (see pall (n.)). Meaning "excuse or extenuate (an offense) by pleading or urging extenuating circumstances or favorable representations" is from 1630s. Related: Palliated; palliating; palliation.
Palliate and extenuate come at essentially the same idea through different figures : palliate is to cover in part as with a cloak; extenuate is to thin away or draw out to fineness. They both refer to the effort to make an offense seem less by bringing forward considerations tending to excuse; they never mean the effort to exonerate or exculpate completely. [Century Dictionary]