How dies Iago use his power of persuasion with Roderigo, Brabantio and Othello?
The easiest answer to your question is that he plays upon their weaknesses.
He knows that if he can incite Brabantio's racial ire -- "an old black ram/Is tupping your white ewe," Brabantio will raise all his power and position against Othello. This might seem an easy feat, but the audience soon learns that Brabantio has been one of Othello's strongest supporters and allies up to learning that Othello has "stolen" his daughter away. Brabantio accuses Othello of using witchcraft to persuade his daughter to marry him.
Iago uses Roderigo's love for Desdemona against him, stringing him along throughout the opening scenes of the play. He has Roderigo convinced that Desdemona will be his if he assists Iago in this way and then that.
And finally, it isn't quite Othello's love for Desdemona that he takes advantage of. Rather, it is Othello's jealousy. It is hard to say why Othello is so available to jealous thoughts about his wife and Cassio, but it could be that Cassio's race and youth prick at Othello's vulnerabilities in these areas.
So, Iago seems to be a master of manipulation of human nature, using all the characters (as well as Emilia and Cassio) to ultimately serve his fell purposes.