Jefferson did not found the Republican Party, Abraham Lincoln did. It was the Republican Party which pushed for abolition of slavery and equal rights. It was the Democrats who wanted to retain slavery. So much for their claims to equality and the party of the common man.
If you are talking about the Democratic Party as of the 1860s, you are right. If you are talking about the Democratic Party today, you are wrong. Same with the GOP.
The Democratic Party between the Civil War and the Great Depression was a party of Southern white male voters. Democratic presidential candidates always won the South, and could win the White House only when they managed to take some of the normally Republican territory (the Northeast, the Midwest, and the West Coast) from them. In order to retain its hold on the South, the Democratic Party adopted positions on racial equality that were moderate at best and outright racist at worst, which made taking those Republican states harder.
That began to change for the Democrats in the 1930s, and completely changed as of the mid-1960s. It began to change for the Republicans in the late 1960s, and has completely changed today.
Today, the Republican party is a party of Southern white male voters. Republican presidential candidates always win the South, and can win the White House only when they manage to take some of the normally Democratic territory (the Northeast, the Midwest, and the West Coast) from them. In order to retain its hold on the South, the Republican Party has abandoned its traditional commitment to civil rights and racial equality, and today adopts positions on these issues that are moderate at best and outright racist at worst, which makes taking those Democratic states harder.
The parties, in short, have completely exchanged places.