I'm certainly not a lawyer but in my personal experience of renting in the past with pets the lease with the pet rent or pet deposit listed in it was the "written permission". I've never been given a separate page stating permission. If it wasn't allowed in the lease the LLs would always cross out the pet section or specifically write in "no pets" in that area. By specifically putting $0 pet deposit and $0 pet rent I, as a tenant, would take that to mean they were aware of the pets (especially if I wrote it on the application, not exactly trying to hide them) and that permission was granted with no extra cost. At least that's how it would sound to me as a tenant.