Line bundles on the moduli stack of parahoric bundles

In this paper we investigate line bundles on $\mathrm{Bun}_{\mathcal{G}}$ the moduli stack of parahoric Bruhat--Tits bundles over a smooth projective curve. Translating this problem into one concerning twisted conformal blocks, we are able to establish criteria that detect when line bundles on an appropriate flag variety descend to $\mathrm{Bun}_{\mathcal{G}}$. Along the way we establish a conjecture of Pappas and Rapoport which describes sections of line bundles on $\mathrm{Bun}_{\mathcal{G}}$ using representation-theoretical means. We conclude the paper with examples where our methods allow us to explicitly determine the Picard group of $\mathrm{Bun}_{\mathcal{G}}$.