We study short crystalline, minimal, essentially self-dual deformations of a mod p non-semisimple Galois representation σ with σ ss = χ k−2 ⊕ ρ ⊕ χ k−1 , where χ is the mod p cyclotomic character and ρ is an absolutely irreducible reduction of the Galois representation ρ f attached to a cusp form f of weight 2k − 2. We show that if the Bloch-Kato Selmer groups H 1 f (Q, ρ f (1 − k) ⊗ Qp/Zp) and H 1 f (Q, ρ(2 − k)) have order p, and there exists a characteristic zero absolutely irreducible deformation of σ then the universal deformation ring is a dvr. When k = 2 this allows us to establish the modularity part of the Paramodular Conjecture in cases when one can find a suitable congruence of Siegel modular forms. As an example we prove the modularity of an abelian surface of conductor 731. When k > 2, we obtain an R red = T theorem showing modularity of all such deformations of σ.