Wait for a Francophone, but it sounds like "I have only one voice". There are many uses for "parole": "tenir la parole" = to have the floor (speak on the floor of Congress, etc.) "tenir ma parole" = to keep my word. "La parole" is speech or the ability to speak. I'm at a loss. Maybe it's "I have only one thing to say".
A formal, or perhaps antiquated, translation would be: "My word is my bond" where bond is a form of legal agreement. It's not something I would use in everyday conversation, but it would be understood by most. I could see it used inside criminal gangs where there is some form of hierarchical structure, such as the Yakuza or Mafia.
And, to invert GerardM's suggestion, another form comes to mind: "I don't go back on my word".