See this documentary here with Stravinsky. Halting the rehearsal, he remarks (at 0:13)
So give me once more the whole period. 39.
According to Wiki
a period consists of two phrases
This section is much longer than two phrases. No way could you do all this in two breaths. Is it/was it an outdated word to use? Period?
The definition of "period" is as you found. It is permissible -- and often necessary -- to breathe during a phrase. In this case, the wind parts are not continuous, so players could sneak small breaths in as needed. However, all of the wind parts I hear would be easily playable in a single breath for a professional musician.
"Period" means many things. One meaning is related to "phrase," but "phrase" also has more than one sense, not necessarily to do with breathing. From Dolmetsch online:
Period: a complete musical thought, concluded by a cadence, having from two phrases, each usually two to eight bars (measures) in length, called the 'antecedent' (often ending in a half cadence) and the 'consequent' (often ending in an authentic cadence)
Phrase: a short musical idea similar to a sentence in spoken language; also a style of performance that gives shape to the musical phrases
Notice the lack of any reference to breath. Players of wind and brass interments, and singers, must consider breath in their phrasing, but this isn't necessarily the same sense of "phrase" that a composer is concerned with. The phrases that make up a period are structural phrases, theoretical or analytical in nature, not necessarily related to breathing.
