It is true that party primaries are generally funded and administered by the state, but the courts have not consistently considered this a sufficient state action nexus for Constitutional limitations to attach. (politics.stackexchange.com)
As you pointed out, parties are private organizations, and therefore are not bound by Constitutional limitations. (politics.stackexchange.com)
However, there are some situations in which Constitutional limitations can apply to private organizations. (politics.stackexchange.com)