No, political parties should not be publicly funded, rather than by donors, because there is no way to administer that system fairly. Anyone could start a policial party and then demand public funding. There is no way to decide who gets how much, or what they can lawfully use it for.
In order to ensure that general fairness of political campaign finance, political parties should be funded by individual donors. Using taxpayers to fund political parties would be forcing politics onto everyone, even those who may not prescribe to a particular political party, or might even disagree with the two party system altogether.