Another thing, AN- you should have defined exactly what you meant by 'influenced.'
I think you could say Republicans have influences like Milton Friedman, Friedrich Hayek, Adam Smith, etc.
Democrats have influences like John Maynard Keynes, Che Guevara, and Marx. You could make the argument that American democrats are influenced by European socialists(who are influenced by Marx) rather than Marx himself.
Keynes was arguably more conservative than so classical liberalsz.
The Republican Party is influenced by Ayn Rand, who is objectively more horrible than Marx (who pretty much all Democrats, btw, have wanted nothing to do with since the 1940s) as well as more influential.
Rand's influence for most people isn't in terms of anything she came up with that was particularly unique, and certainly nothing worth comparing to the lasting historical influence of Marxism.
Though, I don't think Ayn Rand was into opposition to abortion, homosexuality or science. Though, outside of cultural issues, she was very Republican. Very pro-ifyouarenotwithusyouareagainstus and very anti-peoplewhocan'ttakecareofthemselves.