So I used the SuttaCentral search engine to browse through all parimukha references.
They are found equally in AN, SN and MN - less in DN and KN.
Most references don’t mention any meditation method, they just generally describe going into the forest… entering the jhāna. In about another 30% we have parimukha --> hindrances --> jhāna. In the MN this is mostly in the context of describing the gradual training.
And then there are a few suttas with more interesting content relating to our question - and they mostly have parimukha (in the following ‘p’) indeed in connection with ānāpānassati. Most of these are connecting not defining but by sequence, with three exceptions describing ānāpānassati with p. In detail:
AN 3.63 is THE exception. Here we have p --> jhāna, p–> brāmavihāra, p–> understanding I cut off greed, hatred delusion. No mention of ānāpānassati or kayagata sati at all
AN 10.60 is as one of the main meditation suttas of interest. Here we have the 10 saññā, but only ānāpānassati (which is not litereally a saññā) in defining connection with parimukha: “And what, Ānanda, is mindfulness of breathing? Here, a bhikkhu, having gone to the forest, to the foot of a tree…”
SN 54 as the ānāpānasamyutta has in each of the 20 suttas p in connection with ānāpānassati
MN 10 has the many meditations, but again p only before ānāpānassati
MN 119 is the same, p only before ānāpānassati
MN 118 obviously too
MN 62 as another main meditation sutta is peculiar. Rahula sits with parimukha, Sariputta sees him and THEN tells him to do ānāpānassati . Further on more meditations are described, but again, only in connection with ānāpānassati we have p.
DN doesn’t provide interesting details, just DN 22 is as MN 10
KN has a few instances…
Iti 85 is interesting. It has a) ānāpānassati, b) sabbasaṅkhāresu aniccānupassanā and c) asubhānupassanā. p again only in connection with ānāpānassati with the sentence: “ānāpānassati vo ajjhattaṃ parimukhaṃ sūpaṭṭhitā hotu”
“When mindfulness of breathing is inwardly well established parimukham (before one? at the tip of the nose?)” - is the rest an exact translation? I count this is as a defining connection of p and ānāpānassati too.
Ud 7.8 has p with kayagata sati in general, with no mention of ānāpānassati. So that is the second kind-of-exception.
Ps 1.3 again p with ānāpānassati
Finally in the Vinaya we have another direct link of p. and ānāpānassati
Pi Tv Bu Vb Pj 3: “And how is samādhi by mindfulness of breathing developed and cultivated in this way? “A monk sits down in the wilderness, at the foot of a tree, or in an empty hut. He crosses his legs, straightens his body, and establishes mindfulness in front of him”
Conclusion: If parimukha appears in connection with a specific meditation it is in overwhelming majority with ānāpānassati. Sometimes we have other meditations in the same sutta but only ānāpānassati with parimukha (AN 10.60, MN 10, MN 62, MN 119, Iti 85), and two-three times even explaining ānāpānassati with the parimukha passage (AN 10.60, Iti 85, Pi Tv Bu Vb Pj 3).
I didn’t expect this clear connection, but now I think that ānāpānassati actually from the beginning (and not just in Abdhidhamma and commentaries) is described as watching the breath literally ‘around the mouth’ or ‘around the face’ or ‘at the tip of the nose’.
If this is the case, and parimukha would always hint ānāpānassati, then the passages with parimukha that are directly followed by the first jhāna would mean that it was achieved by ānāpānassati.
Open question: has someone already collected the meditations that explicitly lead to the first jhāna other than by ānāpānassati and/or the parimukha passage? (except maybe AN 1.395-574 where almost every dhamma aspect seems to be suitable to develop jhāna…)