One picture holds both uses: a blade kept close to a surface, taking a thin layer off and leaving the body of the thing. What comes away is a peel or a shaving; what stays is the part you wanted. Whether the thing under the blade is an apple, a horse's hoof or next year's budget, the action is the same — go round the outside, take a little at a time, and stop when only what is needed is left.
From the word’s history
Pare came into English through Old French parer 'to arrange, prepare, trim', from Latin parō 'to prepare'. The oldest idea in the word is not damage but getting something ready — trimming a thing into the shape it needs to be.
pare 經古法文 parer（整理、修整）進入英文，更早是拉丁文 parō（準備）。最早的意思不是破壞，而是把東西修整成可用的樣子。這條線一直留到今天：削皮是為了要吃，削預算是為了讓計畫走得下去。
Wiktionary entry 'pare' (kaikki extract, source_ref kaikki:3f6f69a190036ae4:390993722): Middle English paren < Old French parer < Latin parō.
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