To elaborate on user6726's answer, no activities of a political party can be outlawed unless those activities would be illegal even without such membership; or unless those activities would be illegal even if the organization were not a political party.
What would be some examples of when such membership would be punishable by law?
So, for example, an "American Nazi" party would have been illegal during the period of WW2 (from the moment the war was declared until the surrender was signed) because of how treason is defined the US Constitution (article 3.3):
Treason against the United States, shall consist only in levying War against them, or in adhering to their Enemies, giving them Aid and Comfort.
As another example, if any organization (including a political party) were to overtly adopt a platform inciting violence, its members could be reasonably suspected of inciting violence and its leadership would be guilty of inciting violence.
There are some laws which allow for "hate speech" "enhancements" to sentences. Which means that if someone were to be found guilty of committing a violent crime, and it was shown that they engaged in hateful rhetoric, believed to be contributing to their motivation for that crime, then they would receive a harsher prison sentence.
If a "Nazi" party formally adopts a rhetorical platform, which a court agrees to view as hate speech, then a politician running on the platform of that party could potentially receive a harsher sentence if they are convicted of a violent crime or of inciting violence. But such a criminal conviction would have to come first, before sentencing would be even considered.
