Big Pete has been misremembering the order of events, a la Ted in How I Met Your Mother.
Making Pete a straight-up Unreliable Narrator sucks some of the fun out of the series, as it allows us to claim the weirdness of Wellsville is simply Pete making stuff up, and no one wants that. So the idea that Pete's striving to be truthful but has accidentally shuffled some of his stories around allows us to pretend that his relationship with Ellen makes a modicum of sense because it gives us license to reshuffle the romance episodes in any order we want. And after all, Pete's clearly not striving to create a coherent narrative—the series functions more as a series of anecdotes than an autobiography, so he's really got no reason to tell these in any order but the one he wants to. Pete of "Film/Pete's Dragon" is an ancestor of Big Pete and Little Pete.
Think about it- a boy named Pete with bright red hair, a supernatural best friend, and a big imagination in a little town surrounded by cartoonishly eccentric adults? It all fits! The end of the movie is a total parallel to Artie's farewell episode. What if Elliot the dragon knew that he couldn't return because Pete didn't need him anymore, but realized that some day, Pete's great-great grandson would turn out to be a lot like Pete, and he'd need a special friend too? It takes a special kind of child to have their own dragon or superhero- Little Pete's uniqueness turned out to be a family trait, and whatever force that told Elliot he was needed by another kid stepped in to send Artie to Little Pete, because they knew a very similar cycle was about to play out. Artie is on the spectrum, or something.
This was first brought up courtesy of the Nostalgia Critic during his Nickelodeon Month reviews: "Is Artie...special?" This is never addressed on the show, but it does seem odd that Manchild Artie spends all his time with prepubescent kids. Apart from the villainous John Mc Flemp, none of the adults seem concerned about this, nor does Artie give them any reason to be. Also, while Artie is never shown having a typical adult life—i.e., a spouse, a place of his own, a job—he seems like a reasonably competent adult. Ergo, perhaps he has a high-functioning form of autism, or a very mild cognitive disability?
It is a logical inference. There's certainly enough evidence to support it.
