Elizabeth Taylor, accompanied by Alvin the Parrot, planned to put her show on the road this morning for Chicago as "Private Lives" finished its run at the Kennedy Center yesterday.

A new star will rise in the official playbill when the play reopens in Chicago's Shubert Theatre with previews Wednesday. Alvin, at last, will have his own billing (and presumably cooing) in the program.

Alvin was scheduled, as usual, to fly high with his mistress on the plane to Chicago. Fellow travelers say "Alvin is a terror on airplanes." Alvin stands on no ceremony (usually he stands on a convenient shoulder) and talks to strangers after being let out of his cage.

Alvin has taken elocution lessons from the famous actress. He says "I'll get it" when the telephone rings, and then, "hello." Alvin was bought in a pet store in Los Angeles by Taylor. Back then, he expected a life of free caviar with no thought he would ever have to earn his own living.

He accompanied Taylor to New York and did a fly-by for the last two performances. After the applause, there was no keeping the star-struck bird from the footlights. Elizabeth Taylor isn't telling whom he was named for.

In Washington, Alvin was accused by some of having eclipsed Luna, Taylor's fiance', Victor Gonzalez-Luna. But it was Gonzalez-Luna to whom Taylor gave two Givenchy-designed bathrobes with V (for Vista) that came with the Presidential Suite.

Taylor planned to be in Chicago this morning. Burton and his new wife are considering staying in Washington a day longer. Helena Carroll, John Cullum and Charlotte Moore, the New Yorkers in the crowd, hoped to go home for the short interval. Cullum has been flying home most Sunday nights to see his family and then flying back on Tuesdays.

For the Chicago run, everybody has to be in place for rehearsals on the new stage Wednesday afternoon. Sunday is the official opening.

Last night, company manager Michael Lonergan said, six 40-foot trailer trucks pulled up at the stage entrance of the Kennedy Center to load in the unusually architectonic sets.

Contrary to the low spirits when the play came to Washington, they're high after Variety, usually called "the Bible" of the entertainment industry, headlined the Washington stay "A Smash In D.C."

"After the way we had to back out of New York," said Zev Bufman, the play's producer, "Washington was wonderful. We had standing ovations every night. It was very good for Elizabeth. It lifted her spirits. After that, she pumped so much energy into the play and Richard responded, so we've been cutting two or three minutes off the play every night. Ticket sales in Chicago have been phenomenal, as a result of the Washington success."

In her last week in Washington, Elizabeth Taylor turned up in several restaurants off her beaten track. Shirley Kearing, a Washingtonian, said she sat at a table next to Elizabeth Taylor's in Le Fleur Restaurant. Another diner told of seeing Taylor at the West End Cafe' with Al Pacino, also playing the Kennedy Center. The object, one of her favorite restaurateurs was told, was to avoid the crowds that gathered to see her at the Intrigue and Dominique's.

Two bellmen at the Vista International Hotel, who had served her often, gave Taylor violets before she left.