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 (pāna) not breath (pāṇa [Vedic prāṇa]). 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, and while knowing this we train with body, feeling, mind and principles "One Principle (SN 54.1)". 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. What we need to train gets to be the foreground, the centre of our attention, while the meditative inhale and exhale then the background, as context it surrounds that, set to the sides. And this is covered by the word peripheral. With peripherally (parimukhaṃ) then being the adverb.
Pāḷi-English
Parimukhaṃ (adv.) peripherally.