The short answer to this ridiculous question is “no”. Nothing within the popular conception of animal rights, animal liberation, or veganism even suggests that non-human animals should have the rights to vote in elections, get married, run for political office, obtain driver's licenses, or form Super PACs. We believe that all animals, human and non-human alike, have the right to be free from unnecessary suffering and harm. We believe that animals are not our slaves and that we are not their masters and that they have the right to be treated with respect, compassion, and dignity. These are all moral rights, which are distinguished from legal rights, such as the rights to marry and vote. Asserting or implying that the establishment of certain moral rights for non-human animals is either analogous to or will invariably result in the establishment of other unrelated legal rights is either to misunderstand or misrepresent the issue of animal rghts.

Those who are sincerely convinced that the recognition of the moral rights of non-human animals will somehow lead inexorably to the establishment of their legal rights are committing the Slippery Slope fallacy, which occurs when, according to the Skepdic's Dictionary, "[O]ne asserts, without providing any evidence to support the assertion, that an event or chain of events will follow the taking of some action you object to." Of course, there's absolutely no reason to believe that once we human beings arrogate to non-human animals the right to be free from unnecessary harm, it's only a matter of time before cows and chickens are crowding the steps of City Hall to be joined in wedlock.

The Slippery Slope fallacy is a favorite among the opponents of marriage equality, who assert stridently that once same-sex couples are allowed to marry, it's only a matter of time before people are marrying their furniture. As of this writing, marriage equality laws exist in eighteen countries around the world and the number of marriages between a person and a dining table remains zero. 

In the case of deliberately misrepresenting the issue in this way –rather than merely not understanding it properly– it constitutes a classic Straw Man Argument, in which one doesn't confront one's opponent's argument directly, but instead attacks a position that one's opponent does not actually hold and then rips it apart, like a straw man. As the writer Thomas Pynchon once so memorably said, “If they can get you asking the wrong questions, they don't have to worry about the answers.”

Sorry, Scarecrow, but your argument is invalid.