The oldest citations of the word in the OED are from the 15th century, and there is no indication in those where exactly they got the term from back in those days.
Phonetically, it would be somewhat difficult to derive pride from Latin parata. If borrowed straight from Latin, you’d expect it to either simply be a *parata of lions or something like a *parrot of lions (or *parat, or however the spelling might have ended up). If it had gone through French, it would probably have ended up being the same as parry (since Latin -āta gives (Old) French -ée, which was usually borrowed as -y early on). And then there’s the form of parata that went through Italian or Spanish before entering French and thence English: parade. But pride would be quite problematic to derive from parata.
The same, though less insurmountably so, is true of Latin praeda. That certainly could not have gone through French to become English pride, because we already have the French version of praeda in English: prey. The d was regularly lost in French. It could, theoretically, have been borrowed straight from Latin, but equating a stressed ⟨ae⟩ from Latin with a 15th-century (i.e., pre-Great Vowel Shift) English ⟨i⟩ would be quite problematic. Latin ⟨ae⟩ is nearly always [iː] in Modern English, which goes back to an earlier pronunciation [eː] in Middle English. The pronunciation of pride with an [aɪ] diphthong, combined with the consistent spelling with an ⟨i⟩ all the way back to the 15th century, is fairly good evidence that it was in fact an original [iː] sound. And this is not how Latin ⟨ae⟩ was represented.
There is another piece of circumstantial evidence: terms of venery in English are often quite transparently meant to be amusing or exaggerated gibes at perceived characteristics of the animal in question. They are inventions of the 13th and 14th centuries, and when their number and complexity started approaching the ludicrous, so did their silliness. Examples of terms of venery that quite clearly are ‘personifications’, as it were, of perceived characteristics of the animal described (taken from the Wikipedia page linked to by mplungjan in a comment) include:
- a shrewdness of apes/monkeys
- a bellowing of bullfinches (from its rather unpleasant cry)
- a flutter of butterflies
- a pounce of cats (from their tendency to pounce quickly on their prey)
- a peep of chickens
- a chattering/clattering of choughs (from their cry)
- an intrusion of cockroaches
- a bask/float of crocodiles (from their habit of floating near the surface, basking in the sun)
- a murder of crows (seen as a harbinger of death)
- a waddling of ducks
Since lions are—or were, at least, in the Middle Ages—very commonly used as a symbol for pride, bravery, and strength personified (think Richard Lionheart, etc.), it makes good sense to use pride in particular for this perceivedly very proud animal.
So while there may be no way to know with absolute certainty whether the term pride of lions really is a loan from somewhere else,1 I would say that the accumulated evidence in favour of it simply being a perfectly common (in the world of venery terms) use of the regular noun pride is quite a lot more convincing than the evidence that speaks for loan words.
1 Short of there being some obscure old source that actually happens to tell us conclusively of how and when this particular term was first used, and by whom. There may be such a source, but I am unaware of it if there is. The Book of St. Albans does not give any in-depth information, just gives the same blasé mention as the OED and Etymonline, which just shows that already in the 15th century, at least, the term was considered just an extension of the normal word pride, whether that is etymologically justified or not.