Do these private parties use any tax dollars to pay for their primaries? If so I can vote in any of them. I paid for it.
Republicans, Democrats, and even libertarians are in a private party. You have no right voting in their primary to select their candidates for general election. If you truly believe in voting for the man, then your opportunity to make your voice heard is in the general election, choosing between the candidates selected by the members of any parties and independent candidates that may run. Only do what they please liberals really believe they should have a voice in the parties selection of the candidates to present their values and vision.