WHILE it is unusual to find a contemporary Soviet play on a New York stage, it's even more unusual to find a contemporary Soviet play by an author who is not a dissident. Such a rare theatrical specimen is ''The Nest of the Wood Grouse,'' the Viktor Rozov comedy that Joseph Papp has staged at the Public's Newman Theater. Unlike the other Eastern European authors whom Mr. Papp has showcased this year - Vaclav Havel (''A Private View'') and Janusz Glowacki (''Cinders'') - Mr. Rozov co-exists with the authorities at home. As a program insert informs us, ''The Nest of the Wood Grouse'' is ''currently playing to capacity audiences at the Satire Theater in Moscow.''

What kind of play would be a hot ticket in Moscow these days - and be blessed with official tolerance as well? Amusingly enough, ''The Nest of the Wood Grouse'' turns out to be as square and slick - and, in its best passages, as enjoyable and well- acted - as a Broadway domestic comedy of the old school. In fact, Mr. Rozov's play resembles nothing in town so much as ''Brighton Beach Memoirs.'' It's not by chance that Mr. Papp has cast Eli Wallach and Anne Jackson as the parents of Mr. Rozov's contentious household, or that he has commissioned a lavish, symmetrical set (well done by Loren Sherman) that could serve almost any Broadway divertissement designed by Oliver Smith 20 years ago.

The play's title is allegorical: the wood grouse goes deaf during its mating ritual, and, in Russian parlance, the bird's name can refer to someone who is hard of hearing. The extended family in view suffers from all manner of communication failures. Sudakov, the paterfamilias played by Mr. Wallach, is a self-satisfied Foreign Ministry official who revels in the privileges afforded by his rank. Because ''a somebody can do whatever he likes,'' his Moscow apartment is stocked with Oriental rugs, antiques, rare books, icons and artworks picked up during frequent trips abroad. But Sudakov is deaf to the discontent within and beyond his brood. His teen-age son (Ricky Paull Goldin) and married daughter (Mary Beth Hurt) are both in revolt against the inequities of the supposedly classless Soviet society. Even Miss Jackson's stoic mother would like to toss the family's possessions - ''junk,'' she calls them - out the window.

Mr. Rozov's social criticism is mild, if often pungently stated, and it's easy to see why it's state-sanctioned: Officially, at least, the Soviet Government is also against the elitist high-living attacked here. Yet the very tameness of the play's conflicts - parents vs. children, materialism and careerism vs. democratic ideals - makes them wholly accessible to Western audiences. Sudakov's son could well be an American boy who decides to spurn the family plastics business for a career as a social worker.