One thing to think about, I believe, is that ideology ran across party lines more than it ran along them. In the early 1900's, left-parties began to emerge and be successful in Europe, but America lacked that sort of radicalism, though some left-wing ideas were adopted by the progressive movement. The Republicans and Democrats were more tribal or opportunistic unions in some ways than political coalitions, and I think much of the early 20th century showed that the parties' structure no longer matched the ideological structure of the country.

FDR's father was a Democrat, and a Google search seems to claim that Theodore's father was a Republican, but nothing concrete and that may have been all that mattered.