1824 30 Sept. 3/5 Impounded,..a Red Cow, with a white flank, branded on the milking side R.B.
1844 H. Stephens II. 456 There is one side of a cow which is usually called the milking side—that is, the left side.
1891 T. Hardy II. xxiv. 37 She was milking Old Pretty thus, and the sun chancing to be on the milking-side it shone flat upon her pink-gowned form.
1995 (Nexis) 2 Apr. b4 Those who have had the pleasure of performing that early-morning chore have since told her, there's no ‘wrong’ milking side.