Parimukhaṃ is used in the context of ānāpānasati, in the phrase 'parimukhaṃ satiṃ upaṭṭhapetvā', and is a compound with pari (around) and mukha (mouth, entrance). According to the dictionary pari also means '(lit.) away from, off' and mukha 'face, entrance, front, top'.
Ānāpānasati is a compound of ānāpāna and sati. Sati is meditation (see Sati) and ānāpāna is about the breathe not breath. Breath would be pāṇa [Vedic prāṇa] while breathe is pāna, here too the difference in writing concerns just one letter. So in the fourth radiance (jhana) it is the inhaling and exhaling which have stopped, which doesn't mean respiration has stopped; it is not that oxygen is then lacking. Meditation is something we need to develop and here we hone this on the breathe.
We are told to be meditative on the breathe knowing whether it is in or out and f.i. long or short. So, while knowing we train with body, feeling, mind and principles (SN 54.1, SN 54.13). And parimukhaṃ describes this way of attending (upaṭṭhapetvā). We can describe it as 'away from the face', 'away from the front', 'around the front' to indicate what is at the centre of our attention and what at the side. Or at the foreground and background. What we fathom or train gets to be the centre of our attention, the foreground, while the meditative inhale and exhale then surrounds that, as the context or background. And this manner is covered by the word peripheral. With peripherally (parimukhaṃ) then being the adverb.
Pāḷi-English
Parimukhaṃ (adv.) peripherally.