Well, it has a more precise historical meaning.

In nineteenth century Great Britain those who favored the maintenance of the Church of England a.k.a. the Anglican Church as the established or state-sponsored religion were Establishmentarians.

Those who wished to abolish the state-sponsored status of the Church of England were Disestablishmentarians. "Antidistestablishmentarian" was a semi-jocular word for the pro C. of E. faction.