Carol Lempert’s “That Dorothy Parker” is a competently assembled, crisply produced, scrupulously sincere tribute to a major figure in American letters. What on earth is it doing at the New York International Fringe Festival? As countless theatergoers have learned, frequently to their detriment, the Fringe is generally a lot more comfortable with snickers than salutes. Its favorite son remains the Brecht-in-training-pants “Urinetown: The Musical,” and other transfers have included “Debbie Does Dallas,” “Confessions of a Mormon Boy,” and the naughty “Peanuts” homage “Dog Sees God.” But here comes the petite and clear-eyed Ms. Lempert, who has streamlined the life and writings of Parker — the spunkiest and arguably the saddest member of the Algonquin Round Table — into a dot-connecting monodrama that’s about as Fringe-y as “Thurgood” or “Golda’s Balcony.” This isn’t necessarily a bad thing.

After all, Parker appears to have struggled more than her peers with the prospect of being merely glib. She dreaded being remembered as little more than a “wisecracker,” a facile wit who played with words (“Men seldom make passes / At girls who wear glasses”) while the Hemingways and Faulkners wrestled them to the ground and built miraculous new forms.

Ms. Lempert structures her play around the 1943 death of Alexander Woollcott, a self-described “fabulous monster” who (literally) threw his weight around the table. (Ms. Lempert briefly impersonates Woollcott, George S. Kaufman, Peter Benchley, and various other habitués.) Her choice in timing seems strange at first: Parker herself would live another 24 years, with major events such as the Hollywood blacklist still ahead of her. But the framing device provides Ms. Lempert and her director, Janice Goldberg, with an ideal form of vindication: Woollcott and Parker’s long-dormant friendship was rekindled after she shed her droll defenses and wrote movingly of her experiences covering the Spanish Civil War, earning his approval.

Parker once said in a theater review that Katharine Hepburn “ran the whole gamut of emotions from A to B.” Ms. Lempert is safe on that count: She and Ms. Goldberg sketch out an entire lifetime, from bashful girlhood to the thick-footed walk of a middle-age drunkard. The patrician Yankee accent and brittle bonhomie are intact, but Ms. Lempert is not afraid to spotlight Parker’s more wanton, self-destructive behavior as well. (Men do a lot more than make passes here.)

With the exception of the fairly sedate Spanish Civil War section, the play is seasoned with liberal doses of Parker quips, which is something of a mixed blessing. Ms. Lempert’s insights into Parker’s demons and drives are compelling enough that her insistence on shoehorning so many quotable passages into the script — often in half-light as she steps forward — runs the risk of reducing Parker’s inner life to a few couplets of light verse.

And why is it necessary that this vital, tragic woman’s story always be told through the prism of a man? Alan Rudolph’s 1994 biopic, “Mrs. Parker and the Vicious Circle,” framed Parker’s life as a mutually tormented pas de deux with Benchley; Ms. Lempert, meanwhile, presents Woollcott’s approval as the climax of “That Dorothy Parker.” As the Algonquin Round Table continues to shimmer in the minds of actual and honorary New Yorkers alike, it has become increasingly clear that, as far as posterity is concerned, these men need Dorothy Parker a lot more than she needs them.

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The performing began early at Ted Greenberg’s “The Complete Performer,” as several round tables’ worth of wisecracker wannabes held forth outside before being admitted to the theater. Mr. Greenberg won an Emmy Award as a member of David Letterman’s writing staff; were these cutups all fellow comedy writers — or, worse, temps who were convinced that their louder-than-necessary witticisms would be overheard by the right people and result in their becoming fellow comedy writers?

Either way, their efforts proved far more effective once the show started. TV comedy writers thrive off of throwing ideas at one another, and Mr. Greenberg — a trim middle-age fellow who looked like a cross between Larry David and Cal Ripken Jr. — fared best when he engaged the crowd in some mildly hostile banter. Improv suggestions were solicited and then ignored. “Audience awards” offered an excuse to snub a few people in favor of a tall blonde in the front row. The only person to emerge unscathed from his excitable attentions was his personal mascot, who wore a giant foam Ted Greenberg head and led rounds of applause. (“My big furry antidepressant,” the genuine article explained.)

Mr. Greenberg seized his time in the catbird seat with an almost unseemly relish. When it was time for him to hold the stage on his own, however, the wit dipped to the merely diverting. Even at just 30 minutes, his banter suffered from the occasional dry patch, enlivened only when he rejoined the battle against one audience member or another: the woman who called him a narcissist, the man who suggested that his removable turtleneck was less than manly. Until he shows a little bit more of what he’s capable of on his own, Mr. Greenberg would have to be graded as an I for Incomplete.

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Picture the most unoriginal, cliché-prone person you know. Now picture that person falling asleep in front of a Burning Man documentary after bingeing on peyote and Nyquil, and you’ll have some sense of “The Dream-Casting,” a woozy bit of shambolic nonsense courtesy of California’s Teatro Jaguar Luna.

Director/choreographer/guru Huilo Marvavilla aspires to break down the audience’s notions of reality, as his straight-faced cast manipulates eerie masks and periodically ventures off the stage to drape string over the audience’s heads. As the program states, “If you see this event, it is your dream. … Enjoy.” I alone among the smattering of attendees had the inside track on this front: My shadow was inadvertently projected on the lower-left-hand corner of the onstage screen, inserting my head into the psychedelia-by-the-numbers visuals. I did not enjoy.

The reedy Mr. Marvavilla, who closed the program with an exhortation for the audience to audibly “cast” future dreams in unison, let his mellow West Coast vibe slip mid-show when he began berating the sound crew for various flubs. It is possible that the disabling of his microphone was not unintentional, as it came directly before a sequence called “Tea With Duality” in which Mr. Marvavilla gave voice to a pair of hand puppets called Witch Hazel and La Contessa. I have never seen anything like “The Dream-Casting” before; with luck, I won’t again for quite some time.

“That Dorothy Parker” until Sunday (SoHo Playhouse, 15 Vandam St., between Sixth Avenue and Varick Street). “The Complete Performer” until Sunday (Jazz Gallery, 290 Hudson St., between Spring and Dominick streets). “The Dream-Casting” until Friday (Deluxe at Spiegelworld, South Street Seaport). Tickets for all Fringe Festival shows: 866-468-7619.