Why does 'pine feather period' signify the period in a woman's life when she blossoms?

Solution 1:

It's a long-defunct slang usage, resuscitated by OP's citation and the Chicago Artists' Coalition...

Pine Feather Period marks the first of three exhibitions for the sixteen artists selected to participate in Coalition Gallery, a one-year, juried, co-operative exhibition program for emerging, contemporary Chicago artists. In the spirit of a debut, Pine Feather Period, a 1920's flapper slang phrase, describes the period of a debutante's coming out.

As StoneyB points out, it's almost certainly a mistranscription of a trivially figurative usage...

pinfeather - a feather not fully developed; especially : a feather just emerging through the skin.

They're also (dialectally) sometimes penfeathers, but pine feather sounds unlikely to me.

Here's a 1904 example of the usage in approximately OP's sense...

You are now at the Pin-Feather Period, and Mother must teach you how to Fly. I have been giving a lot of Hard Thought to the Man Game [blah blah and I'll help you get successfully married off...]

Solution 2:

Here is the actual entry in Tom Dalzell, Flappers 2 Rappers: American Youth Slang (1996):

pine feather period The period in a Flapper's life when she blossoms out

By way of documenting FumbleFingers's point that "pin-feather period" (not, as typed in Dalzell, "pine feather period") was used during the early decades of the twentieth century to refer to the period of time when a young person is at the stage of development toward independence that corresponds to an unfledged bird, I offer some examples of its use.

Google Books matches for 'pin-feather period'

The earliest Google Books match is from George Ade, "The Coming-Out Girl and a few of Her Keen Guesses," in Current Literature (1904):

"My daughter, we start out for the Country next Week," said the elderly Society Bird to her little Chick. "Us to the Summer Hotel for a bang-up Suite at a Per Diem Rate that will put a large Crimp in Papa's Income for 1904.You are now at the Pin-Feather Period, and Mother must teach you how to Fly. I have been giving a lot of Hard Thought to the Man Game for, lo, these many Moons, and, without passing myself any fragrant Cluster of Green Peas, I think I am On. Every Woman has a private Rogue's Gallery. She can give you a line on the whole Bunko Brotherhood from Sammy the Sophomore, who wears a Buckwheat-Cake instead of a Cap, up to the decrepit old Has-Ben who wants to hold your Hand because you look so much like his daughter.

Another early match is from Richmond Railroad Club, Official Proceedings (1907) [snippet]:

... gain, however, and you are to be congratulated that you escape the speech which I have been “incubating” lo these many days, and really thought that I had successfully pulled through the pin-feather period for a successful flight on the 15th.

And from Carl Werner, "The Pin-Feather Age," in The Outlook (October 18, 1913):

West as a cowboy, and how he had wondered whether he might not get a chance to join the "Bessie, the Beautiful Bowery Maid" company, which was coming to town the next week. Adventure and romance beckoned to him then as they had to me many years before, and as they will to every boy who reaches the pin-feather period.

From an unidentified article in The Silent Partner, volume 10/17 (1914/1921) [combined snippets]:

A man can spread his night sympathy and sentiment all over a side street, but there's no good in gettin' sorry for yourself the next day, just 'cause you feel bum. Getting solid in some society is liable to disturb your permanent position in business.

If you have gone over Fool's Hill, how you will appreciate this article! If you are still in your pin-feather period, you will call it preachin'.

If you fail to advance in your organization, on the pay roll ; if you do not seem to be getting on, just as you think you should, perhaps there's more than preachin' in these observations, by a man who knows that he knows.

And from Emma East, "Mary's Revenge," in Normal Instructor and Primary Plans (March 1921):

"Why a year is a long while," Mary said demurely, "and in that time I have learned the swell dresser's rule o' three—fit, lines, colors. I'm a teacher, you know, and I decided the best Way to exert an influence over the half-baked animals of the pin-feather period was to master the language they liked best, that of nifty clothes, so—"

Aside from two or three literal uses of "pin-feather period" to refer to actual young birds, and a couple of uses of "pin-feather period" in connection with the history of the American Legion (in 1922 and 1923), that's it for the Google Books search results.

Elephind newspaper database matches for 'pin-feather period'

An Elephind search of multiple newspaper databases finds a number of even older matches for the term, used figuratively. From "Brooke!" in the [Logan, Ohio, Hocking] Sent[i]nel (June 18, 1885):

The Yicker-sonians, and the unfledged political chicks, yet in the pin feather period of political life, are tscheaping distressful complaints on account of Brooks' pronounced recognition as the Hocking County Republican leader; but their wails are of no avail. Brooke will possibly nestle them under the protecting wing, hush their chitter with crumbs, or let them starve to silence.

From an untitled item again in the [Logan, Ohio] Hocking Sentinel (May 11, 1893):

Col. McKittrick, the st[or]my petrel, of the Shawnee Journal has sold his paper to Will Shriver, and contemplates going West. We regret the retirement of Bro. McKittrick from the Journalistic field in our neighborhood. He is a brilliant, enterprising dashing writer, grown out of the pin-feather period and of full-fledged wing, strong and defiant in any storm. Our best wishes are for him in what ever clime he may plume his pinions or ply his pencil.

I note here, without quoting the relevant passages, that additional instances of "pin feather period" appear—with the same metaphorical sense of young birdhood—in issues of the Hocking Sentinel from September 12, 1895; April 23, 1896; August 18, 1898; August 22, 1901; and October 22, 1903.

From "Our Tin Wedding," in the *Wichita [Kansas] Daily Eagle (*May 20, 1894):

The Daily EAGLE may be considered as having passed the pin-feather period, this morning being its tenth anniversary. Ten years ago, on the 20th of May, 1884, it essayed its first continuous flight as an Associated Press daily morning paper, in which flight of unwearying wing, bright eye and trusting heart it has never faltered.

From "A New Year and a New Eagle," again in the *Wichita [Kansas] Daily Eagle (*January 1, 1895):

The pin-feather period of the Eagle's ex[is]tence is among the almost forgotten events of the past. For years it has been recognized, far and wide, as a royal fowl of full feather. But the possibilities of a prime maturity were never fully realized.

From "Timely Topics: On the Glorious Fourth, Tribune Wheels and Blue Flame Oil Stoves, in the [Honolulu, Hawaii] Evening Bulletin (June 30, 1896):

An effort is being made to get a scream or two out of the Hawaiian eagle, but it is such a short time since he was hatched out that his scream will not amount to much in comparison with the older bird's [the American Republic], and as for flapping his wings he can't very well do that until they get beyond the pinfeather period.

Conclusions

A couple of things are immediately obvious from these search results. First, metaphorical use of the expression "pin-feather period" is present in the historical record 35 years before the dawn of the 1920s and seems to vanish after 1923. Second, the connection of the expression to young women in the first bloom of youth rests on a single (admittedly very popular) instance from 1903, written by George Ade. The various earlier instances from Ohio, Kansas, and Hawaii apply the expression to newspaper editors and reporters, to newspapers themselves, to amateur poets, to political neophytes, and to a young U.S. territorial possession. And the four instances after Ade refer to the formulating of a speech, a young man in the first bloom of youth, a newcomer to a business, and children in a schoolroom setting.

On the evidence of this limited pool of instances, it appears not only that the phrase "pine feather period" in Flappers 2 Rappers is a typographical error for "pin-feather period," but that the (correctly spelled) expression arose well before the 1920s and was not especially closely tied to the blossoming-out period in a Flapper's life during its brief era of popularity.