Summary: We study the spaces of twisted conformal blocks attached to a \(\Gamma\)-curve \(\Sigma\) with marked \(\Gamma\)-orbits and an action of \(\Gamma\) on a simple Lie algebra \(\mathfrak{g}\), where \(\Gamma\) is a finite group. We prove that if \(\Gamma\) stabilizes a Borel subalgebra of \(\mathfrak{g}\), then the propagation theorem and factorization theorem hold. We endow a flat projective connection on the sheaf of twisted conformal blocks attached to a smooth family of pointed \(\Gamma\)-curves; in particular, it is locally free. We also prove that the sheaf of twisted conformal blocks on the stable compactification of Hurwitz stack is locally free. Let \(\mathscr{G}\) be the parahoric Bruhat-Tits group scheme on the quotient curve \(\Sigma /\Gamma\) obtained via the \(\Gamma\)-invariance of Weil restriction associated to \(\Sigma\) and the simply connected simple algebraic group \(G\) with Lie algebra \(\mathfrak{g}\). We prove that the space of twisted conformal blocks can be identified with the space of generalized theta functions on the moduli stack of quasi-parabolic \(\mathscr{G}\)-torsors on \(\Sigma /\Gamma\) when the level \(c\) is divisible by \(|\Gamma |\) (establishing a conjecture due to Pappas and Rapoport).