If by the opposite of "cool," you mean traditional?

200 years ago, we didn't have Democrat and Republican parties. If we look back to 1815, it was actually the "Era of Good Feelings" post-War of 1812, in which America was able to unite as one and temporarily forget the struggles between the Democratic-Republicans and the Federalists. It's hard to say which would be the "Democrats" and which would be the "Republicans" as they stood for much different things. Federalists were pro-centralized government, while the "Dem-Reps" preferred a decentralized, agrarian-based government.

When you look at the 1850's and 60's, I guess you could say the " traditional conservatives" were the Democrats, as Republican president Abraham Lincoln was the one who freed the slaves. The majority of Democrats, especially Southern Democrats, argued for state's rights and to uphold slavery. Thus the foundation for the ridiculous 21st-century Republican statement that "a Republican freed the slaves" despite the fact that the parties have hardly any resemblance today. Everyone wants Lincoln on their team, it seems.

Okay- now on to 1915. Here's where you can kind of see how the parties are a bit "flip-flopped" from what we know today--here's the Democratic Party platform from 1912, in which it opposes excessive taxation and blames the Republicans for taxing too much.

The high Republican tariff is the principal cause of the unequal distribution of wealth; it is a system of taxation which makes the rich richer and the poor poorer; under its operations the American farmer and laboring man are the chief sufferers; it raises the cost of the necessaries of life to them, but does not protect their product or wages....We favor the immediate downward revision of the existing high and in many cases prohibitive tariff duties, insisting that material reductions be speedily made upon the necessaries of life.

So that's super flipped, obviously. But it seems like the Democrats were always the "trendy environmentalists:"

We believe in the conservation and the development, for the use of all the people, of the natural resources of the country. Our forests, our sources of water supply, our arable and our mineral lands, our navigable streams, and all the other material resources with which our country has been so lavishly endowed, constitute the foundation of our national wealth. Such additional legislation as may be necessary to prevent their being wasted or absorbed by special or privileged interests, should be enacted and the policy of their conservation should be rigidly adhered to.

But keep in mind that Republican president Theodore Roosevelt established many national parks. Source. AND keep in mind that the KKK's second revival in the 1920's had a huge following of Southern Democrats. So, not cool.

TO CONCLUDE: this is very hard to answer because our modern-day parties do not resemble the parties in the past, even if they have the same name. I would say that the "cool" party of 1815 would be the Dem-Reps, as Federalists were on their way out of popularity, eventually to bring about a more permanent two party system.

1860--we are all supposed to say that the Republicans were cool because they were on the right side of the Civil War and freeing the slaves was definitely a great move. So great, that Southerners called Lincoln supporters "Black Republicans" (see the letters of Mary Ann Cobb, Hargrett Special Collections Library, University of Georgia).

And 1915? It was part of the Progressive Era, fighting government corruption which was kinda neat, but we had Republican and Democrat Progressives. I'm sorry if this answer is incomplete, it's just quite hard to try and put this in modern terms.