“There are no small parts, (there are) only small actors” is usually credited to theater director Constantin Stanislavski (1863-1938) of the Moscow Art Theatre. English citations of the Russian use date to 1924, although Stanislavski helped found the Moscow Art Theatre in 1898. The saying means that all roles are important and must be performed well.
 
A Wikipedia entry for the play Our American Cousin suggests that actor Joseph Jefferson used the saying in 1858, but there is no contemporary or reliable documentary evidence to support this.
 
The saying has been associated with New York City’s Metropolitan Opera. Ernestine Schumann-Heink (1861-1938), an operatic contralto, when asked by a New York reporter (see the 1908 citation, below) why she accepted such a small role, said, “There are no small parts in Wagner.” Charles Gilibert, a character baritone at the Met, was credited by at least 1931 for saying “There are no small parts, only small actors.” A 1912 Good Housekeeping article stated, “It is a common axiom behind the scenes that there are no ‘small parts’ in opera.” By at least 1939, the opera saying developed as “There are no small parts, only small singers.”
 
 
Wikipedia: Constantin Stanislavski
Constantin Sergeyevich Stanislavski (Russian: Константин Сергеевич Станиславский) (17 January [O.S. 5 January] 1863 – 7 August 1938), was a Russian actor and theatre director. Building on the directorially-unified aesthetic and ensemble playing of the Meiningen company and the naturalistic staging of Antoine and the independent theatre movement, Stanislavski organized his realistic techniques into a coherent and usable ‘system.’ Thanks to its promotion and development by acting teachers who were former students and the many translations of his theoretical writings, Stanislavski’s ‘system’ acquired an unprecedented ability to cross cultural boundaries and developed an international reach, dominating debates about acting in the West. That many of the precepts of his system seem to be common sense and self-evident testifies to its success.
   
Wikipedia: Our American Cousin
Our American Cousin is an 1858 play in three acts by English playwright Tom Taylor. The play is a farce whose plot is based on the introduction of an awkward, boorish but honest American, Asa Trenchard, to his aristocratic English relatives when he goes to England to claim the family estate. It premiered at Laura Keene’s Theatre in New York City on October 15, 1858, and the main character was first played by Joseph Jefferson. Although the play achieved great renown during its first few years, it became best known as the play U.S. President Abraham Lincoln was attending in Ford’s Theatre when he was assassinated by actor and Confederate sympathizer John Wilkes Booth on April 14, 1865.
 
Theatrical acclaim and “Lord Dun-dreary”
Among Our American Cousin‘s cast was British actor Edward Askew Sothern, playing Lord Dundreary, a caricature of a brainless English nobleman. Sothern had already achieved fame on the New York stage in the play Camille in 1856, and had been reluctant to take on the role, because he felt that it was too small and unimportant. He mentioned his qualms to his friend, Joseph Jefferson, who had been cast in the lead role, and Jefferson supposedly responded with the famous line: “There are no small parts, only small actors.”
 
19 January 1908, Augusta (GA) Chronicle, “The Stage, The Play, The Players,” pg. A5, col. 6:
When questioned by a reviewer as she was leaving the stage of the Metropolitan that evening, he asked her why she had taken such a role for so important an appearance, for it was a small one. Schumann-Heink replied “There are no small parts in Wagner.”
 
13 July 1908, Richmond (VA) Times-Dispatch, pg. 3, cols. 5-6:
A singular fact in connection with the Goodman drama is that there are no small parts or “bits.”
   
Google Books
Good Housekeeping
Volume 54
1912
Pg. ?:
It is a common axiom behind the scenes that there are no “small parts” in opera; meaning by this that every part is important, and that its interpretation must be as perfect as possible.
 
28 December 1924, Canton (OH) Repository, “Director to take lead role in production here,” pg. 18, col. 4:
“There are no small parts, there are only small actors.”
(Dr. John W. Timen, Russian director of the Moscow Art Theater Studies experience—ed.)
 
Google Books
American Hebrew and Jewish Messenger
Volume 116, Issue 22
1925
Pg. 690:
If they only knew, as a famous artist is quoted as saying, that there are no small parts, — there are only small artists.
 
Google Books
Theatre Guild Magazine
Volume 6
1928
Pg. 26:
We remember that he promised us he would write a decalogue for an Acting Company, basing it upon his own experiences with the Moscow Art Theatre. The first commandment, he said, was this: “There are no small parts, only small actors.”
 
Google Books
New Outlook
Volume 157
1931
Pg. 31:
Perhaps she has been reading Stanislavsky, who writes that he used to have painted on the walls of all the dressing rooms of the Moscow Art Theatre the aphorism, “There are no small parts; there are only small actors.”
 
19 November 1931, Dallas (TX) Morning News, “No Small Parts, Only Small Actors on Stage,” sec. 1, pg. 8:
“I (Lawrence Keating—ed.) always recall the remark of the late Charles Gilibert, the great character baritone of the Metropolitan Opera. He said: ‘There are no small parts, only small actors.’”
   
Google Books
The Southern Speech Journal
Volumes 5-8
1939
Pg. 116:
His eye would next fall on a sign above the auditorium door which reads, “There are no small parts, there are only small actors.” This is a quotation from Constantin Stanislavsky, the great Moscow Art Theatre director.
   
Google Books
Opera News
Volume 4
1939
Pg. 97:
Anthony Marlowe does not regret that his voice is of the light buffo variety. He has never forgotten a remark of Alexander Smallens: “There are no small parts, only small singers.”
 
Google Books
Producing Opera in the College
By Louis H. Huber
New York, NY: Bureau of Publications, Teachers College, Columbia
1956
Pg. 36:
Just as “there are no small parts, only small actors,” so there are no small roles in opera, only small singers.
 
Google Books
Recorded Sound
Issues 1-8
1961
Pg. ?:
In this regard, I will again quote Verdi, who declared: ‘There are no small parts — only small singers’.
   
Google Books
Opera
Volume 29, Issues 1-6
1978
Pg. ?:
The supporting roles (I nearly said small, but remembered Vittorio Gui’s remark at a Traviata rehearsal at Covent Garden in 1939 - “there are no small roles in Verdi, only small singers”) were admirably taken, ...
 
Google Books 
Stardust and Shadows:
Canadians in early Hollywood
By Charles Foster
Toronto: Dundurn Press
2000
Pg. 75:
And she (Marie Dressler—ed.) gave a quotation in an interview with a Boston newspaper in 1890, when she was only twenty-one, that has, over the years, been claimed by many famous performers to be their creation. “Remember,” she told the paper she would tell young actors, “there are no small parts, just small actors.”
   
Google Books
The Theatre Quotation Book:
A treasury of insights and insults
Collected and edited by Russell Vandenbroucke
New York, NY: Proscenium Publishers, Inc.
2001
Pg. 91:
There are no small parts, there are only small actors.
KONSTANTIN STANISLAVSKY
 
Google Books
The Complete Idiot’s Guide to Acting
By Paul Baldwin and John Williams Malone
New York, NY: Penguin
2001
Pg. ?:
Small Parts Matter
It is an axiom of the theater that “There are no small parts, only small actors.”