He says hes neutral, but Delphine says she got her dragon info from the very stone slab you you recovered from the barrows... But you gave that to the whiterun mage... not her.

She asked him to retrieve it for her, even the Jarl was involved. She's in there with him before you give the stone to him, just a job that she payed for.

she was there when you gave the stone to Farengar, note this does not happen if you have the stone the first time you enter dragonsreach, plus she was the one who told him where the stone was.