When Lewis says that the Narnia stories are not (strictly speaking) allegories and when Tolkien says that Lord of the Rings is not (in any sense at all) an allegory, they mean different things by the term "allegory".
Lewis limits the use of the term "allegory" to mean a story where concrete objects or characters represent spiritual ideas or internal states. A person hesitating between punching his enemy on the nose and turning the other cheek might be depicted as a jousting tournament between two knights named "Violence" and "Forbearance". Pilgrim's Progress is the most famous example of this kind of writing. (Christian using the key called “hope” to escape from the castle called “Doubt”.)
To the best of my knowledge, no one has ever claimed that either LOTR or Narnia are allegories in this sense. Indeed, I can't off hand think of any modern book that is. Occasionally, you see something a little like it in editorial cartoons in newspapers.
"Parable" means something rather more general, I think, simply "story with a point" or "story with a hidden meaning". The characters in the Biblical parable of the Good Samaritan don't "represent" anything: the whole story illustrates a point: "There’s not much point being religious if you don't do anything about it" and "Well, if your enemy helped you and you countryman ignored you, who would be your neighbor"? Presumably, no one has ever denied that LOTR and Narnia are, or could be read as "parables" in this sense -- that is, they both make moral points. More or less any good story does.
If "parable" means "the story only exists in order to make the point" or "the point came before the story" then I would say that the Narnia books are parables, and the LOTR isn't.
I don't know if there's a word for the third kind of writing, where a fictitious character is used to represent a real one, or an imaginary situation is used as a model or metaphor for some historical situation. Perhaps we could call it a "fable". The obvious example of this would be Animal Farm, where the pigs specifically represent Marx, Lenin, Trotsky, Stalin and so on. At a different level, Lord of the Flies is probably a "fable" in the sense that "the island" represents "the whole world" and the various boys represent "different types of political leader and subject".
When Tolkien vociferously denies that LOTR is an "allegory", he seems to be denying that it is a fable: that is, he is reacting against stupid people who were inclined to say, "The ring is the atomic bomb. The Shire is England. Minas Tirith is the United States. Mordor is Nazi Germany. Isengard is Fascist Italy. Wormtongue is Neville Chamberlain. Eowyn is Vera Lynn." At any rate, in denying that it is "allegory" he points out all the ways in which it does NOT resemble the Second World War.
The school children who wrote to Lewis asking who Reeepicheep "represents" and whether Edmund is Judas seem to have had had the same idea in mind: they had spotted that Aslan was Jesus, concluded that Peter was probably, well, Peter, and assumed that all the other characters must represent a Biblical figure as well.
Rather interestingly, Tolkien, for all he says that he dislikes conscious and intentional "allegory", seems to be quite happy to talk about the Lord of the Rings as if it were an allegory in Lewis's sense. For example
"Elrond symbolizes throughout the ancient wisdom, and his house represents Lore"
"The particular desire of the Eregion elves is an allegory, if you like of the love of machinery and technology..."
He doesn't mind things in the story "representing" states of mind: he does mind them representing specific people and things in the real world.
In "Reflections on the Psalms" Lewis talks about unintentional Second Meanings: a poem or book may "mean" things over and above what the author consciously intended. In this sense, I think that the Lord of the Rings and the Narnia books are full of Second Meanings.
Tolkien did not at any point say, "Gandalf equals Jesus" in the sense that Lewis said, and “Aslan equals Jesus". It would completely mess up the story if you tried to read it that way. But when Gandalf gives up his life in battle with the Balrog, and subsequently rises again in a more obviously divine form, it is impossible for a Christian reader not to think of Jesus. But then, when Frodo is climbing Mount Doom, almost in despair but absolutely committed to completing his mission; with the Ring becoming heavier and heavier, it is hard not to think of Christ's journey to Calvary. And we could also mention Aragorn marching into his City as King and laying hands on people in the Houses of Healing. ("Sing and rejoice for the black gate is broken and your king has passed through and he is victorious.") I don't think that this proves that Tolkien was writing allegory, only that a when a Christian writer tries to imagine a Top Good Guy he necessarily comes up with something Christ like. I'd guess that there were a lot of serene, enlightened heroes in Buddhist fiction.
When Lewis says that Narnia isn't an allegory, I think he is saying that Narnia isn't about "internal states": Peter doesn't represent "faith" or "doubt" or "youth" or "swiss cheese" or anything else: he's just a character. The White Witch doesn't represent "Evil".
But the Narnia stories are also not "fables" in the Animal Farm sense. Everyone agrees that Aslan "is" Jesus. Presumably, the name "Peter" is somewhat intended to make us think of Peter in the Bible -- he's one of Aslan's top helpers and left in charge when Aslan goes away. But that doesn't mean that we expect him to deny Aslan three times, or become bishop of Rome.
I think part of the point of The Lion, the Witch and the Wardrobe is that Lewis's allegories, or analogies, or fables or parables or whatever are not very self-consistent. I think that this is strength, not a weakness, of the books. Lewis doesn't say that Aslan's death at the stone table "is" the crucifixion or "represents" the crucifixion -- he prefers to say that it is "like" it or "intended to make you think of it".
Aslan giving himself over the white witch in return for Edmund's freedom is obviously a picture of Christ’s ransom of the world in a good, old fashioned, penal sense. ("I deserved to die but Christ died in my place.") The American school children were thus completely on the wrong track when they asked if Edmund represented Judas; if he "represents" anything, he represents Adam, or the human race -- the worthless person in whose place the Lion dies.
But the frozen Narnia thawing and the spring returning is ALSO a "picture" of Christ's redemptive death; and then again, so is the sequence in which the animals that the white which has turned into statues come to life. (Lewis specifically compares the New Life in Christ with statues coming to life in "Mere Christianity".) There isn't actually any theology or consistent allegory in LWW, more a series of pictures or illustrations. It must be said that the later books, particularly The Last Battle have much more thought-out religious symbolism and I rather think that the books work less well because of it.
And your point is....? I guess only that it is rather too easy to be dogmatic and say "The Lord of the Rings isn't an allegory and you jolly well mustn't read it like that because Tolkien said so; the Narnia books are Christian parables and you have jolly well got to read them like that because Lewis said so.”
I wonder how we deal with allegory in non-"religious" books? Is Elric an allegory of the non-existence of God? Do children feel violated when they discover that the Wizard of Oz was making subtle secularist points or that Dennis the Menace was filling their heads with subversive anarchism? Or are they less scared of books with ideas in them than their parents think?