@ what the Buddha taught
Parimukhaṃ is used in the context of ānāpānasati. It is found 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. While sati is memory (mindfulness would be sampajañña see SN26.7) it is a bit unfit for the usage in English where recollection fits better. And ānāpāna is about the breathe not breath if that distinction helps. Recollection is something we need to develop and here we hone this on the breathe.
We are told to recollect on the breathe knowing whether it is in or out and long or short, and whilst knowing this train with body, feeling, mind and principles (SN54.1). Parimukhaṃ describes this way of attending (upaṭṭhapetvā). We can describe it as 'away from the front', 'around the front', etc. to indicate what is at the centre of our attention and what is at the side. Or at the foreground and the background. What we train gets to be the centre of our attention, the foreground, and the knowing of the breathe as in or out and long or short then surrounds that, as the context or the background. And this manner is covered by the word peripheral, with peripherally (parimukhaṃ) being the adverb. Recollection of the breathe is a recollection on the breathe but advised is to develop recollection in full (SN54.6). By training with body, feeling, mind and principles while peripherally attending to the recollection on the in and out breathe.
Parimukhaṃ (adv.) [pari+mukha] peripherally.