Fable comes via French from Latin fābula “talk, conversation, gossip or the subject of gossip, a story for entertainment or instruction, a fable.” The plural fābulae is used as an interjection meaning “nonsense! rubbish!”; the idiom lupus in fābulā, literally “the wolf in the fable,” is the equivalent of our “speak of the devil.” The derivative verb fābulārī “to talk, chat” is especially common in the comedies of Plautus and Terence.
Fābulārī, regularized to fābulāre, is the source of Spanish hablar and Portuguese falar “to speak.” Catalan, however, always influenced by French, uses parlar. French parler and Italian parlare are verbs derived from the Latin noun parabola “comparison, explanatory illustration,” in Late Latin (and especially in Christian Latin) “allegorical story, parable, proverb.”
Parabola becomes parola “word” in Italian, parole in French, paraula in Catalan. And by metathesis (transposition of letters) common in Spanish and Portuguese, parabola becomes parabla in Old Spanish, palabra in Spanish, and palavra in Portuguese.
The related English word fib “a small or trivial lie” is a shortening of earlier fibble-fable “nonsense,” an obsolete or dialectal compound based on fable, in the sense “a story not founded in fact.”

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