How do democracies deal with political parties that seek to abrogate democracy?

"Democracies" can't -- by definition they are the will of the people. Yes, some violate people's rights by banning a party or set of speaking points -- but that is a Democracy realizing that they don't really have the tools to legitimately sustain themselves.

A huge improvement, but still not infallible, is a Constitutional Republic / Democracy, where principles can put into writing, and a change requires some kind of change in principle to be encoded into the constitution. Obviously you have to enforce the principles with judges, law enforcement, an informed electorate, etc. Eventually the wrong principles can be encoded, and it is much harder to go back for the same reason it is much harder to change.

So in the end, the best defense a free country with some form of voting and free speech has is not the power to ban anti-democracy forces, but rather an educated populace which can keep educating itself and others on the logical fallacy of a people's party that is anti-people. Hopefully people can learn by logic and concepts, and not by direct experience -- obviously the latter will have learned too late that they were wrong.