Extended Church's thesis and internal parametricity
Internal parametricity does not entail any version of extended Church's thesis. To see this, consider a presheaf model of internal parametricity, for example this one, and observe that in any presheaf model of type theory the extended Church's thesis fails (both the internal and the external one) because the object $\mathbb{N}^\mathbb{N}$ has uncountably many global points.