If a kid went to school he\she cries because he\she is going away from his\her parents not because he\she is going away from his\her home.
So why is it called "homesick" instead of "parents sick" ?
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We don't use "homesick" for a child crying on the first day of school. When people are homesick, it actually is from missing their home. Usually the familiarity of it, the ability to be oneself there, the feeling of safety (either physical, mental, or emotional), or the culture of the area (especially if one is in another country).
If we simply used homesick to describe missing your parents then orphans, the elderly, and those not living with their parents could never be homesick, and obviously they can be.
The historical reason is given here:
homesickness (n.) 1756, translating German Heimweh, from Heim "home" (see home (n.)) + Weh "woe, pain;" the compound is from Swiss dialect, expressing a longing for the mountains, and was introduced to other European languages 17c. by Swiss mercenaries.
It is that way because it expresses a longing for one's home, not one's parents.