Answer. Their fight is with each other, but the court will decide ownership; who signs the storage agreement is not going to carry the day. So in the interest of moving this forward, let them both sign. But it should be under a written agreement saying that: (a) The home has to be inventoried by a neutral third party; (b) Nothing can be removed by one party without advance written notice to and consent of the other; (c) No action can be taken to sell, remove, or dispose of the home without the permission of the court; (d) Each party preserves their claims, but agrees to accept the decision of the court as final.
As for payment of the storage fee, I'd say they should work that out and include it in the writing. They could alternate payments, one pay and the other reimburse half, or any other solution that assures the park getting paid. Then the party who does not get awarded the home would be reimbursed for the storage fees advanced by the party who does.
I suppose it's also possible that the court could require the home be sold prior to close of probate, and then the proceeds divided in whatever way it decides.