Superficially - of course not. Our one and only allegiance is to Hashem and Torah values. Neither the Republican nor the Democratic party look to Torah and Chazal when they form policy.
That being said, look at the issues:
One party is in favor of abortion [that we are forced to pay for] while the other party is in favor of not killing fetuses. The Torah forbids abortion barring very specific and rare conditions being met.
One party is either in favor of the Arabs over the Israelis or views them as having equal rights to sovereignty in Israel while the other is generally strongly Pro-Israel. The Torah is Pro-Israel....
One party wants to strongly restrict the owning of firearms [even for self-defense] while the other is in favor of permitting good people to own firearms. The Torah nowhere forbids one from owning firearms and is quite in favor of self-defense.
One party wants to force the rich to pay huge taxes to support the poor almost to punish them for being rich, to the point where many are Marxists, while the other party believes that while everyone should pay taxes, we shouldn't force socialism on the people. If you made your money fair and square - good for you. The Torah strongly advocates Tzdaka but has no problem with people being rich [unless you want to be really into learning where wealth can be an impediment] and never advocates full blown socialism.
One party favors one part of the population over the other b/c of the color of their skin. The other party believes that we should all be treated equally regardless of skin color. The Torah NEVER says that having a certain type of skin color makes you more or less worthy. Everyone has צלם אלוקים regardless.
One party strongly believes that homosexuality is an equal choice to heterosexuality. Love as you please. Marry whom you please. The other party generally advocates for the man-woman thing. The Torah is quite emphatic in its opposition to homosexuality. Adam and Eve and not Adam and Steve.
One party considers themselves liberal and progressive while the other see themselves as conservative b/c they strive to "conserve" old fashioned values. The Torah is DEFINITELY down with old fashioned values and opposed to contemporary beliefs trumping the timeless and ageless ones we were given at Sinai..
Coincidence? Most Reform Jews, secular Jews and agnostics and atheists of all ethnicities support one party while most religious christians and li-havdil Torah observant Jews support the other.
Since in every instance I mentioned it is the same party whose values conflict with ours and the same party whose values are roughly identical to ours it is a no-brainer which party we should generally be supporting.
This doesn't mean that the politicians of the party we support are all angels. Far from it. It doesn't mean that they are our role models. Not even close. But it does mean that we should support those who promote the values we believe in so deeply.