Psh, the Queen of England doesn't seem all that conservative... but anyway...
I agree, the Democrats were the more conservative of the two parties until the Kennedy era. After that, the meaning of Conservatism was abit more muddled. The 60's were a TERRIBLE time in the 20th century on every front. Sex movements, Vatican II, Decolonisation, etc....
I think true conservatism is experiencing a renaissance since Ron Paul came onto the scene, but we'll see.