Parkinson has guided the League Two side to an improbable appearance in Sunday's Wembley final against Swansea, overcoming three Premier League teams en route to the showpiece.
Wigan, Arsenal and Aston Villa were all stunned by the side from the fourth tier of English football, with Parkinson drawing widespread attention for his achievements.
And Lawn is prepared to offer the Bantams boss and his backroom team the best contract they can afford in a bid to retain his services.
Lawn told The Sun: "It's important we get this deal done and, if we can afford Phil, we'll keep him. We don't just want Phil, we want his backroom staff as well.
"I'm going to lay my cards on the table and offer him as much as I possibly can. I'm not very good at contract negotiations because I just say, 'That's what I've got, I hope it's acceptable'.
"I'm an honest man and I'll say to Phil here's the most I can afford to pay you.
"I won't be messing Phil around, my first offer will genuinely be the best we can do. But Phil says he wants to be here and, as long as that's the case, he'll be here because we want him to stay.
"Give him the right funding and he'll bring you success. He's already proving that."