False, first off Liberal and Conservative is not necessarily merely Republican or Democrat it is a base belief system which constantly is switching between the two parties and on Ideas in general. Conservatism are people a bit more steadfast in their ways cautious of change.
The Republican Party was actually founded by people who were tired of Slavery who came primarily from the Democrat and Wig Parties. They Freed the slaves while the Democratic Party fought tooth and nail to keep them. The Democrats so would go on to be called the Dixiecrat and would found the KKK. The first black people to hold Elected office came from the Republican Party. It is to note that the Democratic Party also elected the Longest Senator in history who incidentally was a Grand Dragoon of the KKK, Mr Byrd.
The first President to even eat dinner with a Black man in office would be Teddy Roosevelt who incidentally was a Republican. It caused great outrage at the time.
The Womans movement was little different, spearheaded By women often belonging to alternative parties it again would be passed by Republicans.
The Civil Rights Legislation would be rammed thru by a Democrat president who absolutely loathed blacks, it would pass nearly completely on a party line of Republican votes with a few northern Democrats siding with it as well.
Democrats passed "Under God" to be placed in your Pledge of Allegiance as well as on your Money.