Actually, your question should be asked the other way around: "when did per become a synonym for a?", since this use of a is much older. And the answer, according to the OED, is around 1400.
The OED says that a was used in this way in Old English, but originaly just for intervals of time. From the OED's citations it seems like it wasn't used for anything but time until the 16th century (sixty tyme a day, twyes a wooke, þries or foure siþes a ȝeere ...).
In Old English this was originally the same word as on, but in late Old English and Early Middle English some uses of on, including this one, lost the /n/.
The first easily comprehensible citation the OED gives is circa 1200:
The sunne..arist anes a dai.
The first citation they give for per is in 1399. And for per, in their first 150 years of citations, it seems only to be used for prices (e.g., russet cloth per yard 15 pence). Here is one from 1413:
I pray and chearge..þat on my stynkyng careyne be neiþer laid cloth of gold ne of silke but russet cloþ per ȝerd xv d.
One more comment ... it appears that, given their respective fields of use, before the 16th century they would only have been synonyms in expressions like "a penny a day".