No, according to Wiktionary[0] the verb parler comes from Early Medieval Latin parabolāre, which comes from Latin parabola (which in Late Latin could have the meaning of "parable"), and that was borrowed from Ancient Greek παραβολή.
The Italian verb parlare has a similar etymology.
edit: it is: https://www.etymonline.com/word/confabulate -- although both the verb fabulari and the noun fabula ultimately come from the proto root bha-, which is also the root for falar in Portuguese and hablar in Spanish (and presumably parlez in French?)