In the Silmarillion, Fëanor's youngest two sons, Amrod and Amras, are twins. Both of their parents are as Elvish as you could want. Fëanor had seven children—a number that was said to be utterly unprecedented for an Elf (before or since)—and this may be a part of that unprecedentedness. (Tolkien never portrayed any other Elf as having more than four children. In the published Silmarillion, Finarfin and his wife had five, but that's the result of Christopher Tolkien simplifying the family tree in a way he later regretted.)
There are a few other sets of Elvish twins, but all the ones I can think of have Men somewhere in their family tree. Beren and Luthien had twin sons (Eluréd and Elurín) as did Elrond and Celebrían (Elladan and Elrohir).
