Summorum Pontificum cura – the third word is the key to the intent of Benedict XVI's Motu Proprio, since the phrase is then complete: "The care of the Supreme Pontiffs [for the sacred liturgy]" – has of course allowed (Art. 9, § 3) that all the clergy may use the 1962 Breviary when fulfilling their mandate of praying the Hours of the Divine Office.
(I assume that if the clergy may use it, certainly a mere layman may; it would be perverse to think otherwise.)
Certain questions arise.
Fairly obviously, Matins is the equivalent of the modern Office of Readings. Now, apparently the clergy under Bishop Rifan in Campos still use (at least the readings from) the pre-1962 Breviary, the main difference being that Pope John XXIII's revision cut down the readings at Matins from nine on all feasts to only three, except for the most important days, and amazingly enough cut out most of the Sunday lessons from the Fathers (thus prioritizing action over contemplation, since the clergy are always busiest on Sunday; a cynic might suggest transferring Sunday's Office to Monday, when most observe their peculiar day of rest...) – this abbreviation is well-critiqued elsewhere.
As mentioned in comments at WDTPRS, presumably this action of the Campos clerics fulfils their obligation, since the greater (all nine readings) includes the lesser (the three remaining after the 1962 abbreviation). But a clarification on this point is needed.
Lauds, Vespers, Compline: old rite or new rite, they're basically the same, just reordered and chopped about (the post-Conciliar edition famously having less psalms per Hour). But what of the Little Hours? Prime has no post-Conciliar equivalent, and of Terce, Sext, and None, tho'all three are still prayable using the options in the new books, only one need be said, hence its name of the Middle Hour (singular). Now, here is the question: One can fulfil one's obligation by praying either Office, and so one can mix-and-match (especially if one prays part of the Office with others, who may use the modern version, while one uses the older in private for the rest of the Office): but how to count the Little Hours?
It would seem, bizarrely enough, that one bound to the Office should either say: Prime, Terce, Sext, and None (1962); or, just one Hour, the Middle Hour (1971). To say only Prime, or Terce, or Sext, or None, or even two or three out of the four, instead of the Middle Hour, would not suffice! Yet, having said Middle Hour, there would be no need to say any of Prime, Terce, Sext, or None! This seems to reverse the principle of the greater including the lesser.
The reason behind this is simple. In either Office, the basic principle is to pray the Psalter over a determinate period of time, disregarding interruptions due to feasts having proper Psalms. In the case of the modern Liturgy of the Hours, the Psalter (omitting about 5% as occasioning psychological difficulties in vernacular recitation – cf. GILH, n. 131) is spread over four weeks, whereas (in line with the counsel of St Benedict and the Fathers) the Roman Breviary spreads the Psalter over one week only. Hence, to fit in all the Psalms, the Breviary must needs have allotted psalms for each of Prime, Terce, Sext, and None, whereas in the Divine Office only one set of psalms is provided for the Middle Hour (those desirous of saying three Hours have to use various of the Gradual Psalms repeatedly, and these are already used at other Hours).
I will be interested to see how this issue is interpreted.
Personally, as a layman I recite the Office out of devotion, realizing that by doing so I am truly taking part in a noble liturgical act, the worship of Christ and His Church, and knowing that liturgical prayer is of its nature higher than non-liturgical, tho' of course many charwomen at their beads please God much more highly than my sinful self distractedly racing through Sext on Sunday.
I used the modern Divine Office for over twelve years, and have just recently switched to the 1962 Breviary; and in both cases I have said the Office complete, not omitting any Hours, but in the case of the modern Office only saying one Middle Hour for the most part (tho' one Lent at least I did say all three of modern Terce, Sext and None, thus becoming very familiar with the Gradual Psalms!). My motive for this is simple: to pray the Office is fundamentally about praising God in His own words, the Psalms (all else is ancillary, tho' very important and holy), and to run through the whole Psalter is the aim; to omit any Hours is to fail in this. So when using the old Breviary, I have said every one of the Little Hours.
It would seem to me wrong to derogate from the discipline and arrangement of the old Breviary to allow the clergy to say only one of the Little Hours from it in fulfilment of their duty; Pope St Pius X intended the cursus to run through all the psalms in the course of each week, and only by saying all the Little Hours can this be achieved; if they esteem it so little, let them pray the "Little Office of Vatican II" as a friend unkindly names the modern Divine Office (which I continue to praise highly, it seems the best of the modern liturgical books), abbreviated precisely for their benefit.