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Your instincts are correct. You win a pair of 20-sided dice.

One of the greatest Elves in the history of Middle-earth, Elrond was born at the Havens of Sirion late in the First Age. During their sack, he and his brother Elros were taken captive by the Sons of Fëanor, but Maglor took pity on them and released them. When they were rescued, Elrond was found playing in a cave beneath a waterfall, and so received his name.

Both of Elrond's parents were half-elven, and after the War of Wrath he was given a choice; whether to be counted amongst the kindred of Elves or of Men. He chose the Firstborn, and remained in Lindon with High King Gil-galad.

After the invasion of Eriador by Sauron in the mid-Second Age, Elrond founded a refuge there in a deep hidden valley; this was named Imladris, or Rivendell in the Common Speech. After Sauron's defeat by Tar-Minastir of Númenor, Elrond remained in Rivendell, where he prospered with the aid of the Great Ring, Vilya.

Elrond played a great part in the histories of the Second and Third Ages; he marched with Gil-galad and Elendil to the War of the Last Alliance, and was a leading member of the White Council. At the time of the Council of Elrond in 3018 (Third Age), he had dwelt in Middle-earth for some 6,500 years.

I once had a fish name Gil-Galad....

Elrond and Arwen weren't pure elves in the first place. I think, I could be wrong though.

No, you're right. Elrond's epithet in the books is "Half-Elven." His daughter is three-quarters elf.

-ccm

After the conclusion of the War of Wrath, Manwë determined that the surviving Half-elven would have their choice of fates: to be counted as Elves, and granted eternal life in the Undying Lands; or to be counted as mortals, and granted the ineffable Gift of Men. This choice could be delayed, although not indefinitely.
Elrond and Arwen are half-elven
Elrond chose to be counted among the Elves, serving in the household of Gil-galad until the end of the Second Age, and founding Rivendell—haven of the Peredhil—in the Third. He married the Elf Celebrían, daughter of Celeborn and Galadriel, and sailed into the West at the conclusion of the War of the Ring.
Arwen chose a mortal life so she could be with Aragon.
marrying Aragorn II Elessar, king of the Reunited Kingdom, and died alone at the age of 2,901 years, grieving the brevity of her mortal happiness. Their son Eldarion and their daughters were not counted as Half-elven, but rather as Dúnedain restored.