I think most people would consider the pronunciation /pəˈrɪfriəl/ "puh-RIF-ri-al" for peripheral nonstandard, for several reasons. Like you, I was unable to find any dictionary that lists it, or any other evidence that it is used by people other than you. It also doesn't match up with the spelling.
The ending -eral is normally pronounced as /ərəl/ or /rəl/). There is generally no stress on the vowel "e" here, and it is often dropped entirely in the pronunciation (examples: general, several, ephemeral, pronounced with /ərəl/ or /rəl/). It generally cannot be pronounced as /əriəl/ or /riəl/.
While a lack of agreement with the spelling doesn't always cause a pronunciation to be considered incorrect (there are words like marriage and carriage where an "i" is written but generally not pronounced), in many cases it does (as in "mischievious" for mischievous, or "grievious" for grievous). I don't remember hearing this pronunciation before, but it seems wrong to me because of this sound-spelling mismatch.
If we do vary the spelling, I found that Oxford Dictionaries does list a variant /ˌpɛrᵻˈfɪərɪəl/ (that's perry-FEAR-ial, so not quite the same pronunciation) that is spelled peripherial. They note it to be rare; the spelling definitely is (see this Ngram, courtesy of Fumblefingers' comment). But apparently, this form actually is older than the form peripheral. So your pronunciation doesn't seem to be archaic, but your inclusion of the -y/-i- before the ending -al is. The adjective peripheral is definitely formed in an unusual way; I haven't been able to find any other adjective ending in -eral that corresponds to a noun ending in -ery. But despite this, it has become the standard form used today.
If we look at the pronunciation of some currently-used words that end in -erial, we can see that they generally display the same stress shift: the adjectives corresponding to artery, ministry, matter and manager are arterial, ministerial, material and managerial, all pronounced with /ˈɪəriəl/.
And expanding to look at other words ending in the sound /riəl/, a vowel letter before it always seems to be stressed, never dropped (see janitorial, adversarial, tenurial). There is a word /ˈfɪmbriəl/ where /riəl/ comes after a consonant (so it has the same phonetic structure as your pronunciation /pəˈrɪfriəl/), but it is spelled fimbrial, not *fimberal (or even *fimberial), and the corresponding noun is fimbria /ˈfɪmbriə/ not *fimbery /ˈfɪmbəri/.
So even if you spell the word peripherial, /pəˈrɪfriəl/ would still be an odd pronunciation. Of course, it's up to you whether you want to change your pronunciation to the more usual one, or stick with your accustomed mumpsimus.
\pə-ˈri-f(ə-)rəl\
and pronounces it pretty much as I would, only a hair stilted.