The word is of biblical origin, but the story is a little strange here. Twice in the Tanach we find the word hora הורה - in Shir HaShirim 3:4 and Hoshea 2:7. In both instances, the word clearly means "mother". And that makes sense - the one who becomes pregnant is indeed the mother.
We do find the word horim, actually horai הורי, in one verse - Bereshit 49:26, in Yaakov's blessing to Yosef:
בִּרְכֹת אָבִיךָ, גָּבְרוּ עַל-בִּרְכֹת הוֹרַי, עַד-תַּאֲוַת, גִּבְעֹת עוֹלָם
The JPS translates this as: "The blessings of your father surpass the blessings of my ancestors [horai], to the utmost bounds of the eternal hills."
However, Sarna comments on the translation "my ancestors":
Hebrew horai is so rendered based on postbiblical usage. However, the stem h-r-h in the Bible can only mean "to become pregnant" and is, of course, solely used in the feminine. Seeing that "mountain(s)" - "hill(s)" is a fixed pair of parallel terms in Hebrew poetry, occuring more than thirty times in that order, Rashbam is undoubtedly correct in connecting horai here with har, "mountain." The Septuagint indeed reads here "ancient mountains," joining the word to the following 'ad. The phrase harere 'ad, "ancient mountains," appears in Habakkuk 3:6 in parallel with give'ot 'olam, "eternal hills." The Blessing of Moses to Joseph in Deuteronomy 33:15 employs the same imagery, though in variant form: "With the best from the ancient mountains, / And the bounty of hills immemorial..." Therefore it is best to render here, "the blessings of the ancient mountains".I find Sarna rather convincing here. Hora certainly seems to mean mother, and the verse in Bereshit does look like it is referring to mountains. So biblically we only had a word referring to the mother, and in post-biblical Hebrew the generic term horeh for "parent" developed. The word "parent" itself in English had a similar development:
from O.Fr. parent (11c.), from L. parentem (nom. parens) "father or mother, ancestor," noun use of prp. of parere "bring forth, give birth to, produce," from PIE base *per- "to bring forth"I haven't seen proof of this, but it wouldn't surprise me if the Latin parentem (or an earlier version) originally meant "mother", since she is the one who gives birth.
What I do find strange is how the words are listed in the dictionary. Both Klein and Even-Shoshan have listings for hora, horeh and horim. Even if they accept the more traditional understanding of Bereshit 49:26 as "my parents", I don't see why they need to have a separate entry for the plural form of the word. I suppose it's to say that the singular horeh means "father", not "parent", whereas only the plural horim is truly generic. But that's not the way it is used in Modern Hebrew. The popular online Hebrew-English dictionary Morfix translates horeh only as "parent", and doesn't have an entry for hora - as mother - at all.
Morfix does, of course, have an entry for another meaning of hora - the folk dance. And no, they're not related. As Philologos discusses here:
“Hora” comes from ancient Greek khoros, which also gives us such words as “chorus” and “choir.” Traditional circle dances deriving their names from khoros can be found all over the Balkans and southeastern Europe. They include the Turkish and Romanian hora, the Bulgarian horo, the Montenegrin and Macedonian ora, and the Russian khorovod, and they are all very old and highly similar in the way they are danced.
And the Online Etymology Dictionary provides the following background to "chorus":
from Gk. khoros "band of dancers or singers, dance, dancing ground," from PIE *ghoro-. In Attic tragedy, the khoros gave expression, between the acts, to the moral and religious sentiments evoked by the actions of the play.
And just in case I have any readers who are accustomed to the Ashkenazi kamatz - there is also no connection to the hora found in the terms "loshon hora" or "yetzer hora". I'll be sure to let you know if I stop using Israeli pronunciation...