It all depends on how it's owned. Whether or not the parent can vote the stock held by a minor child depends on how the stock is owned or held by the minor. Minors can't purchase or open brokerage accounts because they can't enter into legally binding contracts until they reach 18 (or whatever the state law is for majority age). But they can own stock that they have received by gift or inheritance and may have all rights, including voting rights, with regard to that stock. If the minor has the voting rights then the minor could give a parent the proxy to vote those shares (provided proxy voting is allowed).
If you're in California ... If you're incorporated in California, take a look at Cal. Corporations Code Sec. 702 (d) which provides that shares standing in the name of a minor may be voted by proxy unless a guardian of the minor's property has been appointed and written notice of such appointment is given to the corporation.
Another possibility. If the stock is held in a custodial account under the Uniform Gifts to Minors Act or there is some other trust or fiduciary account set up, then the custodian or trustee or other fiduciary would have the voting rights for the stock. As you may be aware, parents often give stock to children under the Uniform Gifts to Minors Act. This allows the donor/parent to give the stock to the minor but the stock is held in a custodial account where the custodian (who can be the donor/parent) controls the account until the child reaches the age of majority. A custodian of stock held in accordance with the Act has sole voting authority over the stock held in the account.