I think it depends on the context. "I have a proof of the BSD conjecture" is correct; maybe you could say "I have proof of the BSD conjecture," but it would sound odd to me. On the other hand, you could say "I have proof that you weren't at the concert last night," but adding "a" would sound odd (to me).You don't need the a here, I have proof of something is correct.
As Outsider says: no "a" here... just "proof."The sentence is
She had a proof of her father killing her mother.